The Devils (14-23-6) tried to rally from 4-0 down with goals Michael McLeod, P.K. Subban and Yegor Sharangovich but lost their fifth game in a row in regulation. Aaron Dell made 24 saves.
The teams will play for a fourth straight time at Prudential Center on Sunday.
"We've talked about this, New Jersey, Buffalo, the teams that are out of it, they're going to play their [tails] off, and if we think it's ever going to be easy we're sadly, sadly mistaken," Quinn said. "I wasn't surprised with the pushback we got from them. … We have a very, very difficult game tomorrow, that's for sure."
Buchnevich gave the Rangers a 1-0 lead at 7:26 of the first period, scoring on a 2-on-1 with Zibanejad.
The Devils committed back-to-back penalties shortly after, with Damon Severson going to the box for interference at 9:06 and a miscommunication among the defensemen leading to a too many men on the ice penalty 46 seconds later.
Ryan Strome scored a 5-on-3 goal at 10:27 to make it 2-0, and Buchnevich scored during a 5-on-4 1:21 later to give New York a 3-0 lead.
"Killed ourselves," Devils coach Lindy Ruff said. "Shot ourselves in the foot. Terrible communication between our 'D', we ended up with too many men on the ice. We came out the way we wanted. We had the energy. We were playing the game the right way. But for me you give that team with that skill a 5-on-3 you're giving them every opportunity to create some momentum."