diff --git a/sanic/app.py b/sanic/app.py
index 163e806d..f0da45b2 100644
--- a/sanic/app.py
+++ b/sanic/app.py
@@ -22,7 +22,7 @@ from sanic.constants import HTTP_METHODS
 from sanic.exceptions import SanicException, ServerError, URLBuildError
 from sanic.handlers import ErrorHandler
 from sanic.log import LOGGING_CONFIG_DEFAULTS, error_logger, logger
-from sanic.response import HTTPResponse, StreamingHTTPResponse
+from sanic.response import BaseHTTPResponse, HTTPResponse
 from sanic.router import Router
 from sanic.server import (
     AsyncioServer,
@@ -1049,10 +1049,8 @@ class Sanic:
                         "Exception occurred in one of response "
                         "middleware handlers"
                     )
-            # Stream response
-            if isinstance(response, StreamingHTTPResponse):
-                await response.stream(request)
-            elif isinstance(response, HTTPResponse):
+            # Make sure that response is finished / run StreamingHTTP callback
+            if isinstance(response, BaseHTTPResponse):
                 await request.respond(response).send(end_stream=True)
             else:
                 raise ServerError(
diff --git a/sanic/response.py b/sanic/response.py
index 2cc74e1e..47f04b39 100644
--- a/sanic/response.py
+++ b/sanic/response.py
@@ -77,6 +77,8 @@ class BaseHTTPResponse:
 
 
 class StreamingHTTPResponse(BaseHTTPResponse):
+    """Old style streaming response. Use `request.respond()` instead of this in
+    new code to avoid the callback."""
     __slots__ = (
         "protocol",
         "streaming_fn",
@@ -105,12 +107,14 @@ class StreamingHTTPResponse(BaseHTTPResponse):
 
         :param data: str or bytes-ish data to be written.
         """
-        await self.send(self._encode_body(data))
+        await super().send(self._encode_body(data))
+
+    async def send(self, *args, **kwargs):
+        if self.streaming_fn is not None:
+            await self.streaming_fn(self)
+            self.streaming_fn = None
+        await super().send(*args, **kwargs)
 
-    async def stream(self, request):
-        request.respond(self)
-        await self.streaming_fn(self)
-        await self.send(end_stream=True)
 
 
 class HTTPResponse(BaseHTTPResponse):