Yes, beforeunload is more reliable, but be sure to assign it directly (not bound through jQuery), like this:
window.onbeforeunload = function() { /* do stuff */ };
The unload event itself wasn’t meant for work to be done, only cleanup of objects…as garbage collectors get better and better, there’s less reason for the browser to even fire the unload event.
Also be aware that for your specific case you’d have to make a synchronous request to the server…otherwise the browser still won’t wait for the AJAX call to complete.