Well, you're required to use the call sign the FCC gave you for legal ID.
That said, I'd say just check in to any nets you're on with the new call when your turn comes around. Depending on how strict and formal the net is you might want to say something short like "formerly KZ7ABC, just got the call for my Extra upgrade"; otherwise, you can let net control or management know when it's convenient. If it's a net that does announcements or formal traffic, you might consider announcing that you just got your new call, or you might send a formal NTS-style message.
The worst that can happen is they'll say they don't have you on the list and will ask for your name and QTH, and then you can tell them "I just upgraded to Extra and got a new call, my old call was (old call) so go ahead and update the net roster." You'll probably get a round of congratulations. Enjoy them, you've earned them.
Oh, and congratulations from me too.