diff --git a/tests/test_connection.py b/tests/test_connection.py
index d8d5f0f..54aeb04 100644
--- a/tests/test_connection.py
+++ b/tests/test_connection.py
@@ -921,3 +921,15 @@ async def handler(request):
     await connection.get_message()
     await connection.aclose()
     await trio.sleep(.1)
+
+
+async def test_finalization_dropped_exception(echo_server, autojump_clock):
+    # Confirm that open_websocket finalization does not contribute to dropped
+    # exceptions as described in https://github.com/python-trio/trio/issues/1559.
+    with pytest.raises(ValueError):
+        with trio.move_on_after(1):
+            async with open_websocket(HOST, echo_server.port, RESOURCE, use_ssl=False):
+                try:
+                    await trio.sleep_forever()
+                finally:
+                    raise ValueError
diff --git a/trio_websocket/_impl.py b/trio_websocket/_impl.py
index a92ce37..3f067ad 100644
--- a/trio_websocket/_impl.py
+++ b/trio_websocket/_impl.py
@@ -1,3 +1,4 @@
+import sys
 from collections import OrderedDict
 from functools import partial
 from ipaddress import ip_address
@@ -35,6 +36,31 @@
 logger = logging.getLogger('trio-websocket')
 
 
+class _preserve_current_exception:
+    """A context manager which should surround an ``__exit__`` or
+    ``__aexit__`` handler or the contents of a ``finally:``
+    block. It ensures that any exception that was being handled
+    upon entry is not masked by a `trio.Cancelled` raised within
+    the body of the context manager.
+
+    https://github.com/python-trio/trio/issues/1559
+    https://gitter.im/python-trio/general?at=5faf2293d37a1a13d6a582cf
+    """
+    __slots__ = ("_armed",)
+
+    def __enter__(self):
+        self._armed = sys.exc_info()[1] is not None
+
+    def __exit__(self, ty, value, tb):
+        if value is None or not self._armed:
+            return False
+
+        def remove_cancels(exc):
+            return None if isinstance(exc, trio.Cancelled) else exc
+
+        return trio.MultiError.filter(remove_cancels, value) is None
+
+
 @asynccontextmanager
 @async_generator
 async def open_websocket(host, port, resource, *, use_ssl, subprotocols=None,
@@ -792,6 +818,10 @@ async def aclose(self, code=1000, reason=None):  # pylint: disable=arguments-dif
         :param int code: A 4-digit code number indicating the type of closure.
         :param str reason: An optional string describing the closure.
         '''
+        with _preserve_current_exception():
+            await self._aclose(code, reason)
+
+    async def _aclose(self, code=1000, reason=None):
         if self._close_reason:
             # Per AsyncResource interface, calling aclose() on a closed resource
             # should succeed.
@@ -964,7 +994,8 @@ async def _close_stream(self):
         ''' Close the TCP connection. '''
         self._reader_running = False
         try:
-            await self._stream.aclose()
+            with _preserve_current_exception():
+                await self._stream.aclose()
         except trio.BrokenResourceError:
             # This means the TCP connection is already dead.
             pass
@@ -1378,6 +1409,6 @@ async def _handle_connection(self, stream):
                 await self._handler(request)
             finally:
                 with trio.move_on_after(self._disconnect_timeout):
-                    # aclose() will shut down the reader task even if its
+                    # aclose() will shut down the reader task even if it's
                     # cancelled:
                     await connection.aclose()