diff --git a/src/cmd/go/internal/work/exec.go b/src/cmd/go/internal/work/exec.go
index 838b00a..a1a357e 100644
--- a/src/cmd/go/internal/work/exec.go
+++ b/src/cmd/go/internal/work/exec.go
@@ -8,6 +8,7 @@
 
 import (
 	"bytes"
+	"cmd/go/internal/fsys"
 	"context"
 	"encoding/json"
 	"errors"
@@ -2242,8 +2243,6 @@
 	// when -trimpath is enabled.
 	if b.gccSupportsFlag(compiler, "-fdebug-prefix-map=a=b") {
 		if cfg.BuildTrimpath {
-			// TODO(#39958): handle overlays
-
 			// Keep in sync with Action.trimpath.
 			// The trimmed paths are a little different, but we need to trim in the
 			// same situations.
@@ -2313,7 +2312,8 @@
 
 	cmdargs := []interface{}{cmd, "-o", outfile, objs, flags}
 	dir := p.Dir
-	out, err := b.runOut(a, dir, b.cCompilerEnv(), cmdargs...)
+	out, err := b.runOut(a, base.Cwd, b.cCompilerEnv(), cmdargs...)
+
 	if len(out) > 0 {
 		// Filter out useless linker warnings caused by bugs outside Go.
 		// See also cmd/link/internal/ld's hostlink method.
@@ -2641,7 +2641,8 @@
 		cgoLDFLAGS = append([]string{"-fsanitize=memory"}, cgoLDFLAGS...)
 	}
 
-	// Allows including _cgo_export.h from .[ch] files in the package.
+	// Allows including _cgo_export.h, as well as the user's .h files,
+	// from .[ch] files in the package.
 	cgoCPPFLAGS = append(cgoCPPFLAGS, "-I", objdir)
 
 	// cgo
@@ -2654,6 +2655,8 @@
 		cfiles = append(cfiles, f+".cgo2.c")
 	}
 
+	hfiles := append([]string{}, p.HFiles...)
+
 	// TODO: make cgo not depend on $GOARCH?
 
 	cgoflags := []string{}
@@ -2698,7 +2701,38 @@
 		cgoflags = append(cgoflags, "-exportheader="+objdir+"_cgo_install.h")
 	}
 
-	if err := b.run(a, p.Dir, p.ImportPath, cgoenv, cfg.BuildToolexec, cgoExe, "-objdir", objdir, "-importpath", p.ImportPath, cgoflags, "--", cgoCPPFLAGS, cgoCFLAGS, cgofiles); err != nil {
+	execdir := p.Dir
+
+	// If any of the Cgo, C, or H files are overlaid, copy them all to
+	// objdir to ensure that they refer to the right header files.
+	// TODO(#39958): Ideally, we'd always do this, but this could
+	// subtly break some cgo files that include .h files across directory
+	// boundaries, even though they shouldn't.
+	hasOverlay := false
+	cgoFileLists := [][]string{cgofiles, gccfiles, gxxfiles, mfiles, ffiles, hfiles}
+OverlayLoop:
+	for _, fs := range cgoFileLists {
+		for _, f := range fs {
+			if _, ok := fsys.OverlayPath(mkAbs(p.Dir, f)); ok {
+				hasOverlay = true
+				break OverlayLoop
+			}
+		}
+	}
+	if hasOverlay {
+		execdir = objdir
+		for _, fs := range cgoFileLists {
+			for i := range fs {
+				opath, _ := fsys.OverlayPath(mkAbs(p.Dir, fs[i]))
+				fs[i] = objdir + filepath.Base(fs[i])
+				if err := b.copyFile(fs[i], opath, 0666, false); err != nil {
+					return nil, nil, err
+				}
+			}
+		}
+	}
+
+	if err := b.run(a, execdir, p.ImportPath, cgoenv, cfg.BuildToolexec, cgoExe, "-objdir", objdir, "-importpath", p.ImportPath, cgoflags, "--", cgoCPPFLAGS, cgoCFLAGS, cgofiles); err != nil {
 		return nil, nil, err
 	}
 	outGo = append(outGo, gofiles...)
@@ -2792,7 +2826,7 @@
 		return err
 	}
 
-	linkobj := str.StringList(ofile, outObj, p.SysoFiles)
+	linkobj := str.StringList(ofile, outObj, mkAbsFiles(p.Dir, p.SysoFiles))
 	dynobj := objdir + "_cgo_.o"
 
 	// we need to use -pie for Linux/ARM to get accurate imported sym
@@ -2817,7 +2851,7 @@
 	if p.Standard && p.ImportPath == "runtime/cgo" {
 		cgoflags = []string{"-dynlinker"} // record path to dynamic linker
 	}
-	return b.run(a, p.Dir, p.ImportPath, b.cCompilerEnv(), cfg.BuildToolexec, cgoExe, "-dynpackage", p.Name, "-dynimport", dynobj, "-dynout", importGo, cgoflags)
+	return b.run(a, base.Cwd, p.ImportPath, b.cCompilerEnv(), cfg.BuildToolexec, cgoExe, "-dynpackage", p.Name, "-dynimport", dynobj, "-dynout", importGo, cgoflags)
 }
 
 // Run SWIG on all SWIG input files.
diff --git a/src/cmd/go/internal/work/gc.go b/src/cmd/go/internal/work/gc.go
index e791734..56711b5 100644
--- a/src/cmd/go/internal/work/gc.go
+++ b/src/cmd/go/internal/work/gc.go
@@ -262,7 +262,7 @@
 	if len(objdir) > 1 && objdir[len(objdir)-1] == filepath.Separator {
 		objdir = objdir[:len(objdir)-1]
 	}
-	rewrite := objdir + "=>"
+	rewrite := ""
 
 	rewriteDir := a.Package.Dir
 	if cfg.BuildTrimpath {
@@ -271,21 +271,54 @@
 		} else {
 			rewriteDir = a.Package.ImportPath
 		}
-		rewrite += ";" + a.Package.Dir + "=>" + rewriteDir
+		rewrite += a.Package.Dir + "=>" + rewriteDir + ";"
 	}
 
 	// Add rewrites for overlays. The 'from' and 'to' paths in overlays don't need to have
 	// same basename, so go from the overlay contents file path (passed to the compiler)
 	// to the path the disk path would be rewritten to.
+
+	cgoFiles := make(map[string]bool)
+	for _, f := range a.Package.CgoFiles {
+		cgoFiles[f] = true
+	}
+
+	// TODO(matloob): Higher up in the stack, when the logic for deciding when to make copies
+	// of c/c++/m/f/hfiles is consolidated, use the same logic that Build uses to determine
+	// whether to create the copies in objdir to decide whether to rewrite objdir to the
+	// package directory here.
+	var overlayNonGoRewrites string // rewrites for non-go files
+	hasCgoOverlay := false
 	if fsys.OverlayFile != "" {
 		for _, filename := range a.Package.AllFiles() {
-			overlayPath, ok := fsys.OverlayPath(filepath.Join(a.Package.Dir, filename))
-			if !ok {
-				continue
+			path := filename
+			if !filepath.IsAbs(path) {
+				path = filepath.Join(a.Package.Dir, path)
 			}
-			rewrite += ";" + overlayPath + "=>" + filepath.Join(rewriteDir, filename)
+			base := filepath.Base(path)
+			isGo := strings.HasSuffix(filename, ".go") || strings.HasSuffix(filename, ".s")
+			isCgo := cgoFiles[filename] || !isGo
+			overlayPath, isOverlay := fsys.OverlayPath(path)
+			if isCgo && isOverlay {
+				hasCgoOverlay = true
+			}
+			if !isCgo && isOverlay {
+				rewrite += overlayPath + "=>" + filepath.Join(rewriteDir, base) + ";"
+			} else if isCgo {
+				// Generate rewrites for non-Go files copied to files in objdir.
+				if filepath.Dir(path) == a.Package.Dir {
+					// This is a file copied to objdir.
+					overlayNonGoRewrites += filepath.Join(objdir, base) + "=>" + filepath.Join(rewriteDir, base) + ";"
+				}
+			} else {
+				// Non-overlay Go files are covered by the a.Package.Dir rewrite rule above.
+			}
 		}
 	}
+	if hasCgoOverlay {
+		rewrite += overlayNonGoRewrites
+	}
+	rewrite += objdir + "=>"
 
 	return rewrite
 }
diff --git a/src/cmd/go/testdata/script/build_overlay.txt b/src/cmd/go/testdata/script/build_overlay.txt
index 0602e70..e18a8f5 100644
--- a/src/cmd/go/testdata/script/build_overlay.txt
+++ b/src/cmd/go/testdata/script/build_overlay.txt
@@ -1,9 +1,11 @@
 [short] skip
 
 # Test building in overlays.
-# TODO(matloob): add a test case where the destination file in the replace map
+# TODO(#39958): add a test case where the destination file in the replace map
 #   isn't a go file. Either completely exclude that case in fs.IsDirWithGoFiles
 #   if the compiler doesn't allow it, or test that it works all the way.
+# TODO(#39958): add a test that both gc and gccgo assembly files can include .h
+#   files.
 
 # The main package (m) is contained in an overlay. It imports m/dir2 which has one
 # file in an overlay and one file outside the overlay, which in turn imports m/dir,
@@ -29,6 +31,18 @@
 stdout $WORK[/\\]gopath[/\\]src[/\\]m[/\\]printpath[/\\]main.go
 stdout $WORK[/\\]gopath[/\\]src[/\\]m[/\\]printpath[/\\]other.go
 
+go build -overlay overlay.json -o main_cgo_replace$GOEXE ./cgo_hello_replace
+exec ./main_cgo_replace$GOEXE
+stdout '^hello cgo\r?\n'
+
+go build -overlay overlay.json -o main_cgo_quote$GOEXE ./cgo_hello_quote
+exec ./main_cgo_quote$GOEXE
+stdout '^hello cgo\r?\n'
+
+go build -overlay overlay.json -o main_cgo_angle$GOEXE ./cgo_hello_angle
+exec ./main_cgo_angle$GOEXE
+stdout '^hello cgo\r?\n'
+
 # Run same tests but with gccgo.
 env GO111MODULE=off
 [!exec:gccgo] stop
@@ -46,6 +60,19 @@
 exec ./print_trimpath_gccgo$GOEXE
 stdout ^\.[/\\]printpath[/\\]main.go
 
+
+go build -compiler=gccgo  -overlay overlay.json -o main_cgo_replace_gccgo$GOEXE ./cgo_hello_replace
+exec ./main_cgo_replace_gccgo$GOEXE
+stdout '^hello cgo\r?\n'
+
+go build -compiler=gccgo  -overlay overlay.json -o main_cgo_quote_gccgo$GOEXE ./cgo_hello_quote
+exec ./main_cgo_quote_gccgo$GOEXE
+stdout '^hello cgo\r?\n'
+
+go build -compiler=gccgo  -overlay overlay.json -o main_cgo_angle_gccgo$GOEXE ./cgo_hello_angle
+exec ./main_cgo_angle_gccgo$GOEXE
+stdout '^hello cgo\r?\n'
+
 -- m/go.mod --
 // TODO(matloob): how do overlays work with go.mod (especially if mod=readonly)
 module m
@@ -71,9 +98,32 @@
 		"dir/g.go": "overlay/dir_g.go",
 		"dir2/i.go": "overlay/dir2_i.go",
 		"printpath/main.go": "overlay/printpath.go",
-		"printpath/other.go": "overlay2/printpath2.go"
+		"printpath/other.go": "overlay2/printpath2.go",
+		"cgo_hello_replace/cgo_header.h": "overlay/cgo_head.h",
+		"cgo_hello_quote/cgo_hello.go": "overlay/cgo_hello_quote.go",
+		"cgo_hello_quote/cgo_header.h": "overlay/cgo_head.h",
+		"cgo_hello_angle/cgo_hello.go": "overlay/cgo_hello_angle.go",
+		"cgo_hello_angle/cgo_header.h": "overlay/cgo_head.h"
 	}
 }
+-- m/cgo_hello_replace/cgo_hello_replace.go --
+package main
+
+// #include "cgo_header.h"
+import "C"
+
+func main() {
+	C.say_hello()
+}
+-- m/cgo_hello_replace/cgo_header.h --
+ // Test that this header is replaced with one that has the proper declaration.
+void say_goodbye();
+
+-- m/cgo_hello_replace/goodbye.c --
+#include <stdio.h>
+
+void say_hello() { puts("hello cgo\n"); fflush(stdout); }
+
 -- m/overlay/f.go --
 package main
 
@@ -128,3 +178,32 @@
 func printMessage() {
 	dir.PrintMessage()
 }
+-- m/overlay/cgo_hello_quote.go --
+package main
+
+// #include "cgo_header.h"
+import "C"
+
+func main() {
+	C.say_hello()
+}
+-- m/overlay/cgo_hello_angle.go --
+package main
+
+// #include <cgo_header.h>
+import "C"
+
+func main() {
+	C.say_hello()
+}
+-- m/overlay/cgo_head.h --
+void say_hello();
+-- m/cgo_hello_quote/hello.c --
+#include <stdio.h>
+
+void say_hello() { puts("hello cgo\n"); fflush(stdout); }
+-- m/cgo_hello_angle/hello.c --
+#include <stdio.h>
+
+void say_hello() { puts("hello cgo\n"); fflush(stdout); }
+
diff --git a/src/cmd/go/testdata/script/build_trimpath_cgo.txt b/src/cmd/go/testdata/script/build_trimpath_cgo.txt
index 4608d9a..3187b4d 100644
--- a/src/cmd/go/testdata/script/build_trimpath_cgo.txt
+++ b/src/cmd/go/testdata/script/build_trimpath_cgo.txt
@@ -20,10 +20,38 @@
 go run ./list-dwarf hello.exe
 ! stdout gopath/src
 
+
+# Do the above, with the cgo (but not .c) sources in an overlay
+# Check that the source path appears when -trimpath is not used.
+mkdir $WORK/overlay
+cp hello.go $WORK/overlay/hello.go
+mkdir hello_overlay
+cp hello.c hello_overlay/hello.c
+go build -overlay overlay.json -o hello_overlay.exe ./hello_overlay
+grep -q gopath[/\\]src hello_overlay.exe
+! grep -q $WORK[/\\]overlay hello_overlay.exe
+go run ./list-dwarf hello_overlay.exe
+stdout gopath[/\\]src
+! stdout $WORK[/\\]overlay
+
+# Check that the source path does not appear when -trimpath is used.
+go build -overlay overlay.json -trimpath -o hello_overlay.exe ./hello_overlay
+! grep -q gopath[/\\]src hello_overlay.exe
+! grep -q $WORK[/\\]overlay hello_overlay.exe
+go run ./list-dwarf hello_overlay.exe
+! stdout gopath/src
+! stdout $WORK[/\\]overlay
+
 -- go.mod --
 module m
 
 go 1.14
+-- overlay.json --
+{
+	"Replace": {
+		"hello_overlay/hello.go": "../../overlay/hello.go"
+	}
+}
 -- hello.c --
 #include <stdio.h>