diff --git a/src/flask/ctx.py b/src/flask/ctx.py
index 8876efd0..849e6c8a 100644
--- a/src/flask/ctx.py
+++ b/src/flask/ctx.py
@@ -484,13 +484,17 @@ class AppContext:
 
         try:
             if self._request is not None:
-                self.app.do_teardown_request(self, exc)
-                self._request.close()
+                try:
+                    self.app.do_teardown_request(self, exc)
+                finally:
+                    self._request.close()
         finally:
-            self.app.do_teardown_appcontext(self, exc)
-            _cv_app.reset(self._cv_token)
-            self._cv_token = None
-            appcontext_popped.send(self.app, _async_wrapper=self.app.ensure_sync)
+            try:
+                self.app.do_teardown_appcontext(self, exc)
+            finally:
+                _cv_app.reset(self._cv_token)
+                self._cv_token = None
+                appcontext_popped.send(self.app, _async_wrapper=self.app.ensure_sync)
 
     def __enter__(self) -> te.Self:
         self.push()