diff --git a/sanic/http/http1.py b/sanic/http/http1.py
index ccfae75d..0cf2fd11 100644
--- a/sanic/http/http1.py
+++ b/sanic/http/http1.py
@@ -16,6 +16,7 @@ from sanic.exceptions import (
     PayloadTooLarge,
     RequestCancelled,
     ServerError,
+    ServiceUnavailable,
 )
 from sanic.headers import format_http1_response
 from sanic.helpers import has_message_body
@@ -428,8 +429,11 @@ class Http(Stream, metaclass=TouchUpMeta):
             if self.request is None:
                 self.create_empty_request()
 
+            request_middleware = not isinstance(exception, ServiceUnavailable)
             try:
-                await app.handle_exception(self.request, exception)
+                await app.handle_exception(
+                    self.request, exception, request_middleware
+                )
             except Exception as e:
                 await app.handle_exception(self.request, e, False)
 
diff --git a/sanic/request.py b/sanic/request.py
index f7b3b999..8b8d9530 100644
--- a/sanic/request.py
+++ b/sanic/request.py
@@ -104,6 +104,7 @@ class Request:
         "_protocol",
         "_remote_addr",
         "_request_middleware_started",
+        "_response_middleware_started",
         "_scheme",
         "_socket",
         "_stream_id",
@@ -179,6 +180,7 @@ class Request:
             Tuple[bool, bool, str, str], List[Tuple[str, str]]
         ] = defaultdict(list)
         self._request_middleware_started = False
+        self._response_middleware_started = False
         self.responded: bool = False
         self.route: Optional[Route] = None
         self.stream: Optional[Stream] = None
@@ -337,7 +339,8 @@ class Request:
             middleware = (
                 self.route and self.route.extra.response_middleware
             ) or self.app.response_middleware
-            if middleware:
+            if middleware and not self._response_middleware_started:
+                self._response_middleware_started = True
                 response = await self.app._run_response_middleware(
                     self, response, middleware
                 )
diff --git a/tests/test_middleware.py b/tests/test_middleware.py
index a2034450..6589f4a4 100644
--- a/tests/test_middleware.py
+++ b/tests/test_middleware.py
@@ -1,6 +1,6 @@
 import logging
 
-from asyncio import CancelledError
+from asyncio import CancelledError, sleep
 from itertools import count
 
 from sanic.exceptions import NotFound
@@ -318,6 +318,32 @@ def test_middleware_return_response(app):
         resp1 = await request.respond()
         return resp1
 
-    _, response = app.test_client.get("/")
+    app.test_client.get("/")
     assert response_middleware_run_count == 1
     assert request_middleware_run_count == 1
+
+
+def test_middleware_run_on_timeout(app):
+    app.config.RESPONSE_TIMEOUT = 0.1
+    response_middleware_run_count = 0
+    request_middleware_run_count = 0
+
+    @app.on_response
+    def response(_, response):
+        nonlocal response_middleware_run_count
+        response_middleware_run_count += 1
+
+    @app.on_request
+    def request(_):
+        nonlocal request_middleware_run_count
+        request_middleware_run_count += 1
+
+    @app.get("/")
+    async def handler(request):
+        resp1 = await request.respond()
+        await sleep(1)
+        return resp1
+
+    app.test_client.get("/")
+    assert request_middleware_run_count == 1
+    assert response_middleware_run_count == 1