diff --git a/call.go b/call.go
index 27cf6411..e4e7771f 100644
--- a/call.go
+++ b/call.go
@@ -51,6 +51,13 @@ import (
 func recvResponse(dopts dialOptions, t transport.ClientTransport, c *callInfo, stream *transport.Stream, reply interface{}) error {
 	// Try to acquire header metadata from the server if there is any.
 	var err error
+	defer func() {
+		if err != nil {
+			if _, ok := err.(transport.ConnectionError); !ok {
+				t.CloseStream(stream, err)
+			}
+		}
+	}()
 	c.headerMD, err = stream.Header()
 	if err != nil {
 		return err
@@ -190,20 +197,18 @@ func Invoke(ctx context.Context, method string, args, reply interface{}, cc *Cli
 			}
 			return toRPCErr(err)
 		}
-		// Receive the response
 		err = recvResponse(cc.dopts, t, &c, stream, reply)
 		if err != nil {
 			if put != nil {
 				put()
 				put = nil
 			}
-			if _, ok := err.(transport.ConnectionError); ok {
+			if _, ok := err.(transport.ConnectionError); ok || err == transport.ErrStreamDrain {
 				if c.failFast {
 					return toRPCErr(err)
 				}
 				continue
 			}
-			t.CloseStream(stream, err)
 			return toRPCErr(err)
 		}
 		if c.traceInfo.tr != nil {
diff --git a/stream.go b/stream.go
index fb7e50f9..008ad1e2 100644
--- a/stream.go
+++ b/stream.go
@@ -165,7 +165,7 @@ func NewClientStream(ctx context.Context, desc *StreamDesc, cc *ClientConn, meth
 				put()
 				put = nil
 			}
-			if _, ok := err.(transport.ConnectionError); ok {
+			if _, ok := err.(transport.ConnectionError); ok || err == transport.ErrStreamDrain {
 				if c.failFast {
 					cs.finish(err)
 					return nil, toRPCErr(err)
diff --git a/test/end2end_test.go b/test/end2end_test.go
index 11e215d7..0bbc3d90 100644
--- a/test/end2end_test.go
+++ b/test/end2end_test.go
@@ -566,12 +566,6 @@ func TestServerGoAway(t *testing.T) {
 func testServerGoAway(t *testing.T, e env) {
 	te := newTest(t, e)
 	te.userAgent = testAppUA
-	te.declareLogNoise(
-		"transport: http2Client.notifyError got notified that the client transport was broken EOF",
-		"grpc: Conn.transportMonitor exits due to: grpc: the client connection is closing",
-		"grpc: Conn.resetTransport failed to create client transport: connection error",
-		"grpc: Conn.resetTransport failed to create client transport: connection error: desc = \"transport: dial unix",
-	)
 	te.startServer(&testServer{security: e.security})
 	defer te.tearDown()
 
diff --git a/transport/http2_client.go b/transport/http2_client.go
index 62a6fa28..6a34c7e5 100644
--- a/transport/http2_client.go
+++ b/transport/http2_client.go
@@ -284,6 +284,10 @@ func (t *http2Client) NewStream(ctx context.Context, callHdr *CallHdr) (_ *Strea
 		t.mu.Unlock()
 		return nil, ErrConnClosing
 	}
+	if t.state == draining {
+		t.mu.Unlock()
+		return nil, ErrStreamDrain
+	}
 	if t.state != reachable {
 		t.mu.Unlock()
 		return nil, ErrConnClosing
@@ -308,6 +312,15 @@ func (t *http2Client) NewStream(ctx context.Context, callHdr *CallHdr) (_ *Strea
 		return nil, err
 	}
 	t.mu.Lock()
+	if t.state == draining {
+		t.mu.Unlock()
+		if checkStreamsQuota {
+			t.streamsQuota.add(1)
+		}
+		// Need to make t writable again so that the rpc in flight can still proceed.
+		t.writableChan <- 0
+		return nil, ErrStreamDrain
+	}
 	if t.state != reachable {
 		t.mu.Unlock()
 		return nil, ErrConnClosing
@@ -419,7 +432,9 @@ func (t *http2Client) CloseStream(s *Stream, err error) {
 	delete(t.activeStreams, s.id)
 	if t.state == draining && len(t.activeStreams) == 0 {
 		// The transport is draining and s is the last live stream on t.
-		defer t.Close()
+		t.mu.Unlock()
+		t.Close()
+		return
 	}
 	t.mu.Unlock()
 	if updateStreams {
@@ -481,7 +496,7 @@ func (t *http2Client) Close() (err error) {
 
 func (t *http2Client) GracefulClose() error {
 	t.mu.Lock()
-	if t.state == closing {
+	if t.state == closing || t.state == unreachable {
 		t.mu.Unlock()
 		return nil
 	}
diff --git a/transport/transport.go b/transport/transport.go
index 86c8fcd6..10b4a2e2 100644
--- a/transport/transport.go
+++ b/transport/transport.go
@@ -240,6 +240,8 @@ func (s *Stream) Header() (metadata.MD, error) {
 	select {
 	case <-s.ctx.Done():
 		return nil, ContextErr(s.ctx.Err())
+	case <-s.goAway:
+		return nil, ErrStreamDrain
 	case <-s.headerChan:
 		return s.header.Copy(), nil
 	}
diff --git a/transport/transport_test.go b/transport/transport_test.go
index 6f9cc50c..f4af68b7 100644
--- a/transport/transport_test.go
+++ b/transport/transport_test.go
@@ -368,8 +368,8 @@ func TestGracefulClose(t *testing.T) {
 		wg.Add(1)
 		go func() {
 			defer wg.Done()
-			if _, err := ct.NewStream(context.Background(), callHdr); err != ErrConnClosing {
-				t.Errorf("%v.NewStream(_, _) = _, %v, want _, %v", ct, err, ErrConnClosing)
+			if _, err := ct.NewStream(context.Background(), callHdr); err != ErrStreamDrain {
+				t.Errorf("%v.NewStream(_, _) = _, %v, want _, %v", ct, err, ErrStreamDrain)
 			}
 		}()
 	}