In general, you can find the logging instance in any library and adjust the logging level

You can access the logging utility via werkzeug._internal

from werkzeug._internal import _logger

then I adjust the root logger to logging.CRITICAL (which is 50)

_logger.setLevel(50)

This means that Werkzeug will only print CRITICAL (or higher) output

You can then adjust the _logger level according got your needs

Direct example

This applies to your example by searching for the import logging command in the Werkzeug github page. This I found in werkzeug._internal then we can do

In [2]: from werkzeug.wrappers import Request, Response
   ...:
   ...: @Request.application
   ...: def application(request):
   ...:     return Response('Hello World!')
   ...:

In [3]: from werkzeug.serving import run_simple

In [4]: run_simple('localhost', 4000, application)
 * Running on http://localhost:4000/ (Press CTRL+C to quit)

In [5]: from werkzeug import _internal

In [6]: _internal._log('info', 'Setting logger level to Critical!')  # see below why this is required
Out[6]: 'Setting logger level to Critical!'

In [7]: _internal._logger.setLevel(50)

In [7]: run_simple('localhost', 4000, application)

In the case the _logger is None then this occurs because no Werkzeug logging instance has been called. See line 75 in _internal for more clarity

In [1]: from werkzeug import _internal

In [2]: type(_internal._logger)
Out[2]: NoneType

In [3]: _internal._log('info', 'removing logger!')
removing logger!

In [4]: type(_internal._logger)
Out[4]: logging.Logger