Mackenzie Blackwood made 17 saves losing his third straight start for the Devils (14-21-6), who have lost seven of eight.
It was the first of a four-game set. They play Thursday and Saturday at Madison Square Garden and Sunday at Prudential Center.
"I think the only disappointing part to me in a game where we skated well and battled hard was some of our net-front battles for second opportunities, our net-front battles to create screens," Devils coach Lindy Ruff said. "We've got to get better at that."
Zibanejad gave the Rangers a 1-0 lead when he skated past defenseman Ryan Murray at the Devils blue line and scored on a snap shot from the slot at 7:02 of the first period.
"When the puck goes in, you get a little more confident," Zibanejad said. "I feel like the puck is going in now and that's nice to be able to contribute."
Shesterkin stopped Jesper Bratt with his right pad on a breakaway attempt 22 seconds into the first and denied Miles Wood with his blocker on a breakaway at 4:16 of the second period.
"He's a great goalie and he keeps being great when we need him," Panarin said. "We need him, and I congratulate him with his first shutout."
The rookie goalie again stopped Wood with his left pad on a shot from the left face-off circle on a 2-on-1 with Nico Hischier at 17:09 of the second.
"If you're going to have success at any point, your goalie has to be good," Quinn said. "He's just calm, knows where the net is, doesn't waste a lot of energy, he's competitive, and it's why he's had so much success at every level."
Hischier, the Devils captain, had two shots on goal in 20:58 in his return to the lineup after missing 24 games with a frontal sinus fracture.
"We've got to get guys [to the net] and we've got to get pucks there and try to get one of those dirty goals," Hischier said. "It's frustrating to not get one in, but I just can't wait to play these guys on Thursday."