diff --git a/src/Servers/Kestrel/Core/src/Internal/Http2/Http2FrameWriter.cs b/src/Servers/Kestrel/Core/src/Internal/Http2/Http2FrameWriter.cs
index 630927bb3ff6..593408337142 100644
--- a/src/Servers/Kestrel/Core/src/Internal/Http2/Http2FrameWriter.cs
+++ b/src/Servers/Kestrel/Core/src/Internal/Http2/Http2FrameWriter.cs
@@ -89,7 +89,6 @@ public void Complete()
 
                 _completed = true;
                 _connectionOutputFlowControl.Abort();
-                _outputWriter.Complete();
             }
         }
 
diff --git a/src/Servers/Kestrel/test/InMemory.FunctionalTests/Http2/Http2TestBase.cs b/src/Servers/Kestrel/test/InMemory.FunctionalTests/Http2/Http2TestBase.cs
index 543a9dc2896b..d9fd7fe6f52f 100644
--- a/src/Servers/Kestrel/test/InMemory.FunctionalTests/Http2/Http2TestBase.cs
+++ b/src/Servers/Kestrel/test/InMemory.FunctionalTests/Http2/Http2TestBase.cs
@@ -467,7 +467,22 @@ protected async Task InitializeConnectionAsync(RequestDelegate application, int
                 CreateConnection();
             }
 
-            _connectionTask = _connection.ProcessRequestsAsync(new DummyApplication(application));
+            var connectionTask = _connection.ProcessRequestsAsync(new DummyApplication(application));
+
+            async Task CompletePipeOnTaskCompletion()
+            {
+                try
+                {
+                    await connectionTask;
+                }
+                finally
+                {
+                    _pair.Transport.Input.Complete();
+                    _pair.Transport.Output.Complete();
+                }
+            }
+
+            _connectionTask = CompletePipeOnTaskCompletion();
 
             await SendPreambleAsync().ConfigureAwait(false);
             await SendSettingsAsync();