Jerry Stackhouse landed a three-pointer with less than four minutes remaining in overtime to level matters, and the Nets pushed on to clinch victory in a game where neither team led by more than seven.
It was the first meeting between the teams in Brooklyn, and the victory for the Nets puts them joint-top with their city rivals.
This game was originally scheduled as the season opener but was postponed due to Hurricane Sandy.
"Having two great teams in New York, the fans are in the best position of all," said Nets centre Brook Lopez, who had 22 points and 11 rebounds.
Deron Williams scored 16 points and had 14 assists as the Nets improved their home record to 7-1.
The Knicks were without injured trio Jason Kidd, Amar'e Stoudemire and Iman Shumpert, but Brooklyn-born Carmelo Anthony shouldered the scoring responsibility by finishing with 35 points.
"I guess you can say this is the start of something," said Anthony.
"To step back from this and take my Knicks uniform off and be a Brooklyn kid once again, to be a part of that atmosphere, there was nothing like it."