Brook Lopez had 28 points and 10 rebounds while Deron Williams collected 23 points and 10 assists in a do-or-die game for Brooklyn.
They used a late 15-1 run to kill off the Bulls, who were led by 20 points and eight assists from Nate Robinson.
The Atlanta Hawks hung on for a 102-91 victory over the Indiana Pacers that levels their series at 2-2.
Josh Smith poured in 29 points as the Hawks went 17 up at half-time and then held off the Pacers to take the win.
The Houston Rockets staved off elimination with a thrilling 105-103 victory over Oklahoma City.
Having squandered late leads in the previous two games, the Rockets watched the Thunder twice come up empty with shots that would have tied it in the final seconds.
Chandler Parsons led Houston with 27 points, while Kevin Durant had 38 in a losing cause for the Thunder.