diff --git a/sanic/app.py b/sanic/app.py
index f0da45b2..c48a8d20 100644
--- a/sanic/app.py
+++ b/sanic/app.py
@@ -938,7 +938,7 @@ class Sanic:
         """
         pass
 
-    async def handle_exception(self, request, exception, name=""):
+    async def handle_exception(self, request, exception):
         try:
             response = self.error_handler.response(request, exception)
             if isawaitable(response):
@@ -959,7 +959,7 @@ class Sanic:
         if response is not None:
             try:
                 response = await self._run_response_middleware(
-                    request, response, request_name=name
+                    request, response, request_name=request.name
                 )
             except CancelledError:
                 # FIXME: Ensure exiting in a clean manner instead of this
@@ -992,6 +992,7 @@ class Sanic:
         try:
             # Fetch handler from router
             handler, args, kwargs, uri, name = self.router.get(request)
+            request.name = name
 
             if request.stream.request_body:
                 if self.router.is_stream_handler(request):
@@ -1036,22 +1037,11 @@ class Sanic:
                 response = handler(request, *args, **kwargs)
                 if isawaitable(response):
                     response = await response
-            # Run response middleware
-            if response is not None:
-                try:
-                    response = await self._run_response_middleware(
-                        request, response, request_name=name
-                    )
-                except CancelledError:
-                    raise
-                except Exception:
-                    error_logger.exception(
-                        "Exception occurred in one of response "
-                        "middleware handlers"
-                    )
+            if response:
+                response = await request.respond(response)
             # Make sure that response is finished / run StreamingHTTP callback
             if isinstance(response, BaseHTTPResponse):
-                await request.respond(response).send(end_stream=True)
+                await response.send(end_stream=True)
             else:
                 raise ServerError(
                     f"Invalid response type {response!r} (need HTTPResponse)"
@@ -1063,8 +1053,9 @@ class Sanic:
             # -------------------------------------------- #
             # Response Generation Failed
             # -------------------------------------------- #
-            response = await self.handle_exception(request, e, name)
-            await request.respond(response).send(end_stream=True)
+            response = await self.handle_exception(request, e)
+            response = await request.respond(response)
+            await response.send(end_stream=True)
 
     # -------------------------------------------------------------------- #
     # Testing
@@ -1345,6 +1336,8 @@ class Sanic:
                     _response = await _response
                 if _response:
                     response = _response
+                    if isinstance(response, BaseHTTPResponse):
+                        response = request.stream.respond(response)
                     break
         return response
 
diff --git a/sanic/compat.py b/sanic/compat.py
index ebf25bb0..e6d0bc31 100644
--- a/sanic/compat.py
+++ b/sanic/compat.py
@@ -1,7 +1,13 @@
 from sys import argv
 
 from multidict import CIMultiDict  # type: ignore
+from asyncio import CancelledError
 
+try:
+    from trio import Cancelled
+    CancelledErrors = CancelledError, Cancelled
+except ImportError:
+    CancelledErrors = CancelledError,
 
 class Header(CIMultiDict):
     def get_all(self, key):
diff --git a/sanic/request.py b/sanic/request.py
index fb98cbae..ce9e1b51 100644
--- a/sanic/request.py
+++ b/sanic/request.py
@@ -9,6 +9,7 @@ from urllib.parse import parse_qs, parse_qsl, unquote, urlunparse
 from httptools import parse_url  # type: ignore
 
 from sanic.exceptions import InvalidUsage
+from sanic.compat import CancelledErrors
 from sanic.headers import (
     parse_content_header,
     parse_forwarded,
@@ -16,7 +17,7 @@ from sanic.headers import (
     parse_xforwarded,
 )
 from sanic.log import error_logger, logger
-from sanic.response import HTTPResponse
+from sanic.response import HTTPResponse, BaseHTTPResponse
 
 
 try:
@@ -62,6 +63,7 @@ class Request:
         "endpoint",
         "headers",
         "method",
+        "name",
         "parsed_args",
         "parsed_not_grouped_args",
         "parsed_files",
@@ -89,6 +91,7 @@ class Request:
         # Init but do not inhale
         self.body = b""
         self.ctx = SimpleNamespace()
+        self.name = None
         self.parsed_forwarded = None
         self.parsed_json = None
         self.parsed_form = None
@@ -105,7 +108,7 @@ class Request:
             self.__class__.__name__, self.method, self.path
         )
 
-    def respond(
+    async def respond(
         self, response=None, *, status=200, headers=None, content_type=None
     ):
         # This logic of determining which response to use is subject to change
@@ -113,8 +116,22 @@ class Request:
             response = self.stream.response or HTTPResponse(
                 status=status, headers=headers, content_type=content_type,
             )
-        # Connect the response and return it
-        return self.stream.respond(response)
+        # Connect the response
+        if isinstance(response, BaseHTTPResponse):
+            response = self.stream.respond(response)
+        # Run response middleware
+        try:
+            response = await self.app._run_response_middleware(
+                self, response, request_name=self.name
+            )
+        except CancelledErrors:
+            raise
+        except Exception:
+            error_logger.exception(
+                "Exception occurred in one of response "
+                "middleware handlers"
+            )
+        return response
 
     async def receive_body(self):
         self.body = b"".join([data async for data in self.stream])