diff --git a/src/cmd/go/internal/modload/query.go b/src/cmd/go/internal/modload/query.go
index 602bf47275..f3c003a7cd 100644
--- a/src/cmd/go/internal/modload/query.go
+++ b/src/cmd/go/internal/modload/query.go
@@ -471,7 +471,17 @@ func queryPrefixModules(candidateModules []string, queryModule func(path string)
 					notExistErr = rErr
 				}
 			} else if err == nil {
-				err = r.err
+				if len(found) > 0 {
+					// golang.org/issue/34094: If we have already found a module
+					// containing the target package, ignore errors for modules with
+					// shorter paths.
+
+					// golang.org/issue/34383 is a special case of this: if we have
+					// already found example.com/foo/v2@v2.0.0 with a matching go.mod
+					// file, ignore the error from example.com/foo@v2.0.0.
+				} else {
+					err = r.err
+				}
 			}
 		}
 	}
diff --git a/src/cmd/go/testdata/script/mod_get_major.txt b/src/cmd/go/testdata/script/mod_get_major.txt
new file mode 100644
index 0000000000..dd1cbe01a3
--- /dev/null
+++ b/src/cmd/go/testdata/script/mod_get_major.txt
@@ -0,0 +1,18 @@
+[!net] skip
+[!exec:git] skip
+
+env GO111MODULE=on
+env GOPROXY=direct
+env GOSUMDB=off
+
+# golang.org/issue/34383: if a module path ends in a major-version suffix,
+# ensure that 'direct' mode can resolve the package to a module.
+
+go get -d vcs-test.golang.org/git/v3pkg.git/v3@v3.0.0
+
+go list -m vcs-test.golang.org/git/v3pkg.git/v3
+stdout '^vcs-test.golang.org/git/v3pkg.git/v3 v3.0.0$'
+
+-- go.mod --
+module example.com
+go 1.13