Green connected on 11-of-18 shot from the field. Deron Williams added 18 points and distributed 10 assists for the Nets, who won for the fifth time in seven contests.
Kris Humphries had 16 points and 11 rebounds for the winners.
The Cavaliers fell for the 10th time in 11 games despite Antawn Jamison's 34 points.
Lester Hudson scored a career-high 26 points off the bench, but it wasn't enough for Cleveland.