Lawson Crouse and Nick Bjugstad scored, Dylan Guenther had two assists, and Connor Ingram made 36 saves for Arizona (2-5-1), which has lost four of its past five games, including its home opener, 3-2 in overtime to the Winnipeg Jets on Friday.
"[New York is] one of the best teams in the League, and we stuck around with them for 55 minutes," Ingram said. "We didn't get the result we wanted, but it's a step forward, a step in the right direction, to show we can be competitive in this league."
Lafreniere gave the Rangers a 1-0 lead at 1:31 of the second period, scoring off a pass from Panarin as he was crashing the net.
"When you play with players like that, you just go to the net and things happen, and he found me," Lafreniere said. "It was a great play by him."
Crouse tied it 1-1 on a power play at 6:45, finishing off a tic-tac-toe passing sequence with Guenther and Matias Maccelli.
Bjugstad then gave the Coyotes a 2-1 lead at 11:27 with a backhand after Rangers defenseman Ryan Lindgren couldn't clear the puck in front.
"Ingram played unbelievable for us," Bjugstad said. "We had our chances. We have to take the positives from it and keep growing as a team. That's a really good team over there, but obviously we would have liked to have had this one."
Panarin tied it 2-2 at 18:06, holding off Coyotes defenseman Shayne Gostisbehere for a wrist shot that deflected in off Ingram's left shoulder.
"We've got some high-end players, and when we play well, they come through for us," Gallant said. "What I liked about our game tonight was it was everybody, the third line, the fourth line, the so-called third pair defense. I thought they all played their role real well."
Zibanejad put the Rangers back in front 3-2 at 12:56 of the third period, scoring backdoor on a power play off a touch pass from Chris Kreider.
"It was a great play [by Panarin] to get it to 'Kreids,' and for me, I'm just trying to hit the net and get the puck to go in," Zibanejad said.
