DeMarre Carroll tied the game with a three-pointer at the buzzer in the second overtime, before D'Angelo Russell scored 14 of his 36 points in the third to edge the Nets ahead.
Eight Brooklyn players finished in double figures, giving them a 30-29 record heading into the All-Star break.
An impressive 42 points from James Harden was not enough to lift the Houston Rockets over the Minnesota Timberwolves.
Harden posted a 31st consecutive game with 30 points or more to move into joint second place for the second-longest streak in NBA history.
But strong defensive play from Josh Okogie helped ensure the Timberwolves triumphed 121-111 and pick up their second straight win.
— Timberwolves (@Timberwolves) February 14, 2019
Despite trailing heading into the fourth quarter, the Portland Trail Blazers surged ahead to win 129-107 against the Golden State Warriors.
An impressive block by Zach Collins with 7:35 left to play helped shift the momentum in Portland's favour, and they outscored the Warriors 25-4 from there.
Eight Trail Blazers finished in double figures, led by Damian Lillard on 29.
Meanwhile, the Toronto Raptors secured their sixth consecutive victory with a 129-120 defeat of the Washington Wizards.
Pascal Siakam led the way with a career-high 44 points as the Raptors surged in the third quarter to rally from 12 points behind.
Nikola Jokic's bucket with less than a second left lifted the Denver Nuggets to a 120-118 victory against the Sacramento Kings.
And the Philadelphia 76ers put Tuesday's loss to the Boston Celtics behind them as they notched up a 126-111 win over the New York Knicks.
The Phoenix Suns were resigned to their 15th straight loss, falling 134-107 to the Los Angeles Clippers.
Elsewhere, Giannis Antetokounmpo had a triple-double and the Milwaukee Bucks came back from a 10-point deficit in the fourth quarter to beat the Indiana Pacers 106-97, while Dwayne Wade had 22 points as the Miami Heat defeated the Dallas Mavericks 112-101.
Otto Porter Jr recorded a career-high 37 points in the Chicago Bulls' 122-110 win over the Memphis Grizzlies, and the Boston Celtics head into the All-Star break with back-to-back wins after outlasting the Detroit Pistons 118-110.