diff --git a/src/cmd/go/internal/test/test.go b/src/cmd/go/internal/test/test.go
index 0ac56666201..5a8cc1e8c12 100644
--- a/src/cmd/go/internal/test/test.go
+++ b/src/cmd/go/internal/test/test.go
@@ -858,7 +858,10 @@ func runTest(ctx context.Context, cmd *base.Command, args []string) {
 	if cfg.BuildCoverPkg != nil {
 		match := make([]func(*load.Package) bool, len(cfg.BuildCoverPkg))
 		for i := range cfg.BuildCoverPkg {
-			match[i] = load.MatchPackage(cfg.BuildCoverPkg[i], base.Cwd())
+			match[i] = func(p *load.Package) bool {
+				cwd := base.Cwd()
+				return load.MatchPackage(cfg.BuildCoverPkg[i], cwd)(p) && !isUnderSpecial(cwd, p.Dir)
+			}
 		}
 
 		// Select for coverage all dependencies matching the -coverpkg
@@ -2112,3 +2115,36 @@ func testBinaryName(p *load.Package) string {
 
 	return elem + ".test"
 }
+
+// isUnderSpecial checks whether dir is contained within a 'special' directory under 'cwd'.
+// A directory is special if it beings with "." or "_" , or is called "testdata"
+func isUnderSpecial(cwd string, dir string) bool {
+	rel, err := filepath.Rel(cwd, dir)
+	if err != nil {
+		return false
+	}
+
+	hasAnyPrefix := func(dir string, prefixes ...string) bool {
+		for _, prefix := range prefixes {
+			if strings.HasPrefix(dir, prefix) {
+				return true
+			}
+		}
+		return false
+	}
+
+	sep := string(filepath.Separator)
+	if rel == "." || hasAnyPrefix(rel, ".."+sep) {
+		// Not a special directory under 'cwd', so can return immediately
+		return false
+	}
+
+	// Otherwise avoid special directories "testdata" or beginning with ".", "_".
+	pathComponents := strings.Split(rel, sep)
+	for _, elem := range pathComponents {
+		if hasAnyPrefix(elem, ".", "_") || elem == "testdata" {
+			return true
+		}
+	}
+	return false
+}
diff --git a/src/cmd/go/testdata/script/coverpkg_ignores_special_dirs.txt b/src/cmd/go/testdata/script/coverpkg_ignores_special_dirs.txt
new file mode 100644
index 00000000000..9ce3cbc4c4c
--- /dev/null
+++ b/src/cmd/go/testdata/script/coverpkg_ignores_special_dirs.txt
@@ -0,0 +1,57 @@
+# test for https://github.com/golang/go/issues/66038
+[short] skip
+
+env GO111MODULE=off
+# files
+env GOPATH=$WORK/.dir${:}$WORK/_dir${:}$WORK/testdata
+
+cd $WORK
+go test -coverpkg=./... ./...
+stdout 'coverage: 100.0%'
+
+-- $WORK/a.go --
+package a
+
+// trivial function with 100% test coverage
+
+import (
+	_ "dot_dir"
+	_ "under_dir"
+	_ "testdata_dir"
+)
+
+func F(i int) int {
+	return i*i
+}
+
+-- $WORK/a_test.go --
+package a
+
+import (
+	"testing"
+)
+
+func TestF(t *testing.T) {
+	F(2)
+}
+
+-- $WORK/.dir/src/dot_dir/b.go --
+package dot_dir
+
+func G(i int) int {
+	return i*i
+}
+
+-- $WORK/_dir/src/under_dir/b.go --
+package dot_dir
+
+func G(i int) int {
+	return i*i
+}
+
+-- $WORK/testdata/src/testdata_dir/b.go --
+package testdata_dir
+
+func G(i int) int {
+	return i*i
+}