diff --git a/asyncpg/connect_utils.py b/asyncpg/connect_utils.py
index 2678b358..c2f0901b 100644
--- a/asyncpg/connect_utils.py
+++ b/asyncpg/connect_utils.py
@@ -626,12 +626,23 @@ async def _connect_addr(*, addr, loop, timeout, params, config,
     else:
         connector = loop.create_connection(proto_factory, *addr)
 
+    def _close_leaked_connection(fut):
+        if not fut.cancelled():
+            tr, pr = fut.result()
+            if tr:
+                tr.close()
+
     connector = asyncio.ensure_future(connector)
     before = time.monotonic()
     try:
         tr, pr = await asyncio.wait_for(
             connector, timeout=timeout)
     except asyncio.CancelledError:
+        # Ensure connection is closed.
+        # The cancellation may have raced the connect, leading
+        # to the connect completing but the wait_for to be
+        # cancelled.
+        # See: https://bugs.python.org/issue37658
         connector.add_done_callback(_close_leaked_connection)
         raise
     timeout -= time.monotonic() - before
@@ -732,12 +743,3 @@ def _create_future(loop):
         return asyncio.Future(loop=loop)
     else:
         return create_future()
-
-
-def _close_leaked_connection(fut):
-    try:
-        tr, pr = fut.result()
-        if tr:
-            tr.close()
-    except asyncio.CancelledError:
-        pass  # hide the exception
diff --git a/asyncpg/pool.py b/asyncpg/pool.py
index 20a3234e..ed9ccff8 100644
--- a/asyncpg/pool.py
+++ b/asyncpg/pool.py
@@ -605,15 +605,28 @@ async def _acquire_impl():
                 ch._timeout = timeout
                 return proxy
 
+        def _release_leaked_connection(fut):
+            if not fut.cancelled():
+                asyncio.ensure_future(self.release(fut.result()))
+
         if self._closing:
             raise exceptions.InterfaceError('pool is closing')
         self._check_init()
 
-        if timeout is None:
-            return await _acquire_impl()
-        else:
+        acquire_fut = asyncio.ensure_future(_acquire_impl())
+        try:
+            # Calling wait_for with timeout=None will shortcut to run without
+            # timeout
             return await asyncio.wait_for(
-                _acquire_impl(), timeout=timeout)
+                acquire_fut, timeout=timeout)
+        except asyncio.CancelledError:
+            # Ensure connection is marked as not in use.
+            # The cancellation may have raced the acquire, leading
+            # to the acquire completing but the wait_for to be
+            # cancelled.
+            # See: https://bugs.python.org/issue37658
+            acquire_fut.add_done_callback(_release_leaked_connection)
+            raise
 
     async def release(self, connection, *, timeout=None):
         """Release a database connection back to the pool.
diff --git a/tests/test_pool.py b/tests/test_pool.py
index e51923e4..9857dceb 100644
--- a/tests/test_pool.py
+++ b/tests/test_pool.py
@@ -379,6 +379,26 @@ async def worker():
             self.cluster.trust_local_connections()
             self.cluster.reload()
 
+    async def test_pool_handles_task_cancel_in_acquire_with_timeout(self):
+        # See https://github.com/MagicStack/asyncpg/issues/547
+        pool = await self.create_pool(database='postgres',
+                                      min_size=1, max_size=1)
+
+        async def worker():
+            async with pool.acquire(timeout=100):
+                pass
+
+        # Schedule task
+        task = self.loop.create_task(worker())
+        # Yield to task, but cancel almost immediately
+        await asyncio.sleep(0.00000000001)
+        # Cancel the worker.
+        task.cancel()
+        # Wait to make sure the cleanup has completed.
+        await asyncio.sleep(0.4)
+        # Check that the connection has been returned to the pool.
+        self.assertEqual(pool._queue.qsize(), 1)
+
     async def test_pool_handles_task_cancel_in_release(self):
         # Use SlowResetConnectionPool to simulate
         # the Task.cancel() and __aexit__ race.