The Rangers (31-35-9) previously lost to New Jersey 6-3 on March 7 and by the same score on March 18.
Connor Sheary scored for a second straight game, and J.T. Miller, Jaroslav Chmelar and Mika Zibanejad also scored. Adam Fox had two assists. In addition to the saves, Shesterkin fought Devils goalie Jacob Markstrom at 9:05 of the third period, his first NHL fight.
The Rangers have already been eliminated from postseason contention.
The Devils (38-34-2), who began the night 10 points out of a wild-card position in the Eastern Conference, moved closer to elimination with their second loss in three games and third in six.
Connor Brown scored and Markstrom made 19 saves.
Sheary gave New York a 1-0 lead at 8:14 of the first period when from the right face-off circle he deflected Tye Kartye’s shot from the right point down and past Markstrom with Miller battling with Jonas Siegenthaler in front of the goalie.
The Rangers extended the lead to 2-0 with Miller’s power-play goal at 13:39.
Zibanejad spun the puck back to Fox at the left point. Fox delivered a pass to Alexis Lafreniere at the bottom of the right circle. Lafreniere one-timed it to the front where Miller was positioned for a quick redirection into the net.
Brown scored on the power play 51 seconds into the second period for the Devils to make it 2-1.
Chmelar scored off his own rebound at 12:12 to give the Rangers a 3-1 lead. He deflected Braden Schneider’s shot off Markstrom’s pads, got to the rebound on the left side and whacked it in with his backhand.
Zibanejad extended New York’s lead to 4-1 at 3:49 of the third period. He dashed to the front of the net, getting behind Jack Hughes and Siegenthaler, and redirected Gabe Perreault’s pass past Markstrom.
