Green's slam off a feed from Andre Iguodala with 59 seconds left in the extra session gave the Warriors a 106-104 lead.
Stephen Curry then sealed the outcome with a jumper with 13.5 seconds remaining, giving the Warriors a fifth consecutive victory and sixth straight against the Cavaliers.
Curry provided 29 points, 11 assists and nine rebounds.
David Lee added 19 points for Golden State.
Cleveland got 27 points and nine assists from Kyrie Irving but couldn't avoid its fifth straight loss.
Tristan Thompson logged 17 points and 12 rebounds for the Cavs.