If you are storing your cart as a session, then some languages allow
you to give a session an identifier. I know PHP allows this. In that
case, you would simply pass the session id to google as merchange
private data and then when your site get the new order notification,
look up the session and kill its contents. Not sure if you can do this
with other languages.
By and large, however, it is better to put carts in a database so that
it is persistent. Customers like going back to a site x days later and
seeing their items still in their cart. Using sessions would have
resulted in the cart expiring.
Speaking of which, if your only concern is killing the cart, why not
just let the cart die when the session expires?