Had the exat same problem with my two Moto G's I ordered from Amazon.de. It seems to happen to some people after the update.
Calm down, no panic. Both of my phones still work flawlessly and battery life is same as before if not better.
So how did I fix it? I factory reseted the Moto G (probably not necessary) and then I let the battery drain to dead (from whatever % it had until it shut down). Most comfortable way to do this is probably playing, streaming videos as I did.
Somehow the update messed around with the battery calibration.
I hope I've helped you