Theme div.item-list of views pager?
Typically, unless you are using a specific contributed module, the pager is a template file that comes with views. You can override theme_pager for the specific view (or all your paged views) and add the additional div.

https://api.drupal.org/api/drupal/includes!pager.inc/function/theme_pager/7

Examples on how to do this:
https://www.drupal.org/node/387786