How can I Prevent CSRF Tokens from being Cached? We have a Craft site in which we include a partial that includes in several different places. CSRF is enabled, and we're appending {{ getCsrfInput() }} to the form.

Because of the way we've set up the site, and are including this partial in so many places, nearly all of the forms we're rendering are using cached CSRF tokens, which invalidates them.

Is there a way to prevent Craft's {% cache %} tag from including the generated CSRF token without ending the cache and starting it up again each time? (eg, {% endcache %}{{ getCsrfInpt() }}{% cache %}) 

I worry doing so would be risky, because we may include this partial in a part of the site that is not cached, so there'd be no cache tag to end.

Thanks!

Update: I failed to mention that our site works right now without Javascript, so I'd like to avoid it. Otherwise, Lindsey's answer is perfect.
You can do something like this:

// First, set all your cached content to a variable:
{% set myContent %}
  {% cache %}
    {# YOUR CACHED CONTENT #}
    %%MY_CSRF%%
  {% endcache %}
{% endset %}

// Then, run a replace on your cached content to insert the new CSRF
{{ myContent|replace({'%%MY_CSRF%%': getCsrfInput()}) }}