The Warriors owned a 19-point lead in the third period but collapsed down the stretch while being outscored in the fourth quarter 32-16.
Green then scored on a dunk to tie it again and, after a Golden State turnover, made the layup that gave the Nets their first lead since the opening 2 minutes of the game.
A missed layup by Golden State's Brandon Rush with 35 seconds to go and a failed jump shot attempt by Charles Jenkins with 2 seconds remaining left Golden State with its fourth straight loss.
New Jersey won its second in a row in the opener of a four-game road trip.
Gerald Wallace had 24 points and 18 rebounds for the Nets. Green and Humphries added 20 each.
Golden State was led by Lee, who made 12-of-16 from the floor and tallied 27 points.