You can try doing an AJAX type callback in the OnUnload event - however, as someone else mentioned you'd have to be aware of other tabs being open (not easy), and it still won't guarantee you get that event.
There's a combination of things to do to get a similar type of effect.
Session Cookie should have a null/empty expiry time. This ensures the browser deletes the session from it's end after the browser is closed.
The ASP Session can be set with a short SessionState timeout value. This means if there's no client activity within that period, the Session will expire.
The side effect of this is that if a user is just looking at the site, and not performing activity (regardless of whether the browser is still open) - the session can expire.
This can be worked-around by having a Javascript timer periodically ping back to the server with an AJAX call or similar. The side effect of THIS is that it generates more site traffic.