There is no postaback, like in standard asp.net, there can be only form tag that posts data to some url (controller/action).
Inside your partial user control, write:
<form action="controller/actionname" method="post">
<input type="text" name="inputText" />
<input type="submit" value="Post data to server" />
</form>
In MVC, only input type="submit" triggers form submit. In standard ASP.NET webforms, you can have many Linkbuttons, Buttons, ... but under cover, they all triggers this simple click on input type="submit" through javascript event. One form can post data to only one URL (controller/action), but that can be changed with javascript (as we can see in html source of 'old' asp.net webforms).
then, in controller you can handle post data:
[AcceptVerb(HttpVerb.Post)] // optionally
public ActionResult ActionName(string inputText) ...