In East Rutherford, N.J., Cleveland's Edgar Jones tipped in a missed shot with four seconds left to beat New Jersey.
The Nets, who lost their fourth consecutive game, tied the game with 14 seconds remaining on Darwin Cook's three-point shot from the top of the key.
After a timeout, Roy Hinson missed a shot, rebounded and missed again for the Cavaliers. Jones then tapped the ball in over Mike Gminski.
The victory was only the Cavaliers' second in the last nine games as they fight for the eighth and final Eastern Conference playoff spot.
New Jersey has lost nine of its last 12 games. Combined with Washington's victory over Dallas, the loss dropped the Nets into the conference's seventh playoff spot with eight games remaining.
The Nets trailed, 90-78, after John Bagley's 70-foot basket at the third-quarter buzzer for the Cavaliers. Midway in the final period, New Jersey outscored Cleveland by 11-2, with Cook getting four points, to pull within 102-101 with 3:12 remaining.
Hinson hit a foul shot for the Cavaliers, but Albert King's 19-footer tied it at 103.
World B. Free, who led all scorers with 26 points, then hit a jumper and Jones added a foul shot for a 106-103 Cleveland lead with 1:35 to go.
King, who had 19 points, and Hinson traded jump shots before Cook's three-pointer.
Free led all scorers with 26 points. New Jersey was led by Gminski with 24 and Mickey Johnson with 20