diff --git a/playwright/_impl/_browser_type.py b/playwright/_impl/_browser_type.py
index dcc8db413..0e664a27d 100644
--- a/playwright/_impl/_browser_type.py
+++ b/playwright/_impl/_browser_type.py
@@ -257,15 +257,8 @@ def handle_transport_close(reason: Optional[str]) -> None:
                     for page in context.pages:
                         page._on_close()
                     context._on_close()
-            connection.cleanup(reason)
-            # Give a chance to any API call promises to reject upon page/context closure.
-            # This happens naturally when we receive page.onClose and browser.onClose from the server
-            # in separate tasks. However, upon pipe closure we used to dispatch them all synchronously
-            # here and promises did not have a chance to reject.
-            # The order of rejects vs closure is a part of the API contract and our test runner
-            # relies on it to attribute rejections to the right test.
-            if browser:
                 connection._loop.call_soon(browser._on_close)
+            connection.cleanup(reason)
 
         transport.once("close", handle_transport_close)
 
diff --git a/playwright/_impl/_connection.py b/playwright/_impl/_connection.py
index 57a9dcf6d..29f781f49 100644
--- a/playwright/_impl/_connection.py
+++ b/playwright/_impl/_connection.py
@@ -144,13 +144,7 @@ async def _inner_send(
             self._object, method, augmented_params, timeout
         )
         try:
-            done, _ = await asyncio.wait(
-                {
-                    self._connection._transport.on_error_future,
-                    callback.future,
-                },
-                return_when=asyncio.FIRST_COMPLETED,
-            )
+            result = await callback.future
         except asyncio.CancelledError as exc:
             await self._connection._abort(
                 self._object,
@@ -158,9 +152,6 @@ async def _inner_send(
                 str(exc) or "Task was cancelled",
             )
             raise
-        if not callback.future.done():
-            callback.future.cancel()
-        result = next(iter(done)).result()
         # Protocol now has named return values, assume result is one level deeper unless
         # there is explicit ambiguity.
         if not result:
@@ -351,9 +342,20 @@ async def init() -> None:
                 if not self.playwright_future.done():
                     self.playwright_future.set_exception(exc)
 
-        await self._transport.connect()
-        self._init_task = self._loop.create_task(init())
-        await self._transport.run()
+        try:
+            await self._transport.connect()
+            self._init_task = self._loop.create_task(init())
+            await self._transport.run()
+        finally:
+            cause = None
+            if (
+                self._transport.on_error_future.done()
+                and not self._transport.on_error_future.cancelled()
+            ):
+                transport_exc = self._transport.on_error_future.exception()
+                if transport_exc is not None:
+                    cause = str(transport_exc)
+            self.cleanup(cause)
 
     def stop_sync(self) -> None:
         self._transport.request_stop()
@@ -367,6 +369,8 @@ async def stop_async(self) -> None:
         self.cleanup()
 
     def cleanup(self, cause: str = None) -> None:
+        if self._closed_error:
+            return
         self._closed_error = TargetClosedError(cause) if cause else TargetClosedError()
         if self._init_task and not self._init_task.done():
             self._init_task.cancel()
@@ -463,19 +467,12 @@ async def _abort(
         except (Error, OSError):
             pass
         try:
-            done, _ = await asyncio.wait(
-                {
-                    self._transport.on_error_future,
-                    callback.future,
-                },
-                return_when=asyncio.FIRST_COMPLETED,
-            )
+            await callback.future
+        except (Exception, asyncio.CancelledError):
+            pass
         finally:
             if not callback.future.done():
                 callback.future.cancel()
-        for future in done:
-            if not future.cancelled():
-                future.exception()
 
     def dispatch(self, msg: ParsedMessagePayload) -> None:
         if self._closed_error: