New York took a 2-1 lead on Zibanejad's 39th goal of the season at 7:18.
"I don't think we were very good in the first [period]," Rangers defenseman Marc Staal said. "We just weren't playing well. We were getting beat to pucks. We were playing way too loose."
It continued in the second period, when the Devils scored four goals.
"We were ready to put our push on and play our game," Palmieri said.
Palmieri scored his second of the game at 44 seconds to tie it 2-2. Claesson scored his first with the Devils 52 seconds later for a 3-2 lead. The defenseman was acquired in a trade with the Carolina Hurricanes on Feb. 24.
Zajac scored on the power play at 16:19 to extend the lead to 4-2 and Hayden scored 25 seconds later on a deflection to make it 5-2.
"We made it real easy for them," Zibanejad said. "We just didn't have an answer. We dug ourselves a hole. Going forward, with the playoff-type of games we have in front of us, we can't do that. … Just a bad performance overall."
Greg McKegg scored on a deflection at 13:42 of the third period to make it 5-3. Zajac's empty-net goal at 19:04 extended the lead to 6-3, and Tony DeAngelo scored at 19:17 for the 6-4 final.
"We owed them," Zajac said. "We've gotten beat by them [two times] this year. We owed them one. It's an important time for them in the season so it makes it sweeter for us"