Zibanejad scored on a wrist shot from the high slot at 17:00 to give the Rangers a 4-3 lead.
New York (23-16-6), 5-0-1 in its past six games, swept a four-game series to finish the season 6-2-0 against New Jersey.
The Rangers went 2-for-6 on the power play and have scored nine goals with the man-advantage in their past nine games.
"I think we move the puck and we attack (on the power play)," Zibanejad said. "I think we play at a higher tempo, moving the puck and supporting each other. We work hard to get the puck back, and we're scoring right now, so that's huge for our team."
The Devils (14-24-6) rallied from down 3-0 to tie it on goals by Mikhail Maltsev, Marián Studenič and Nico Hischier but lost their sixth game in a row, all in regulation. Mackenzie Blackwood made 21 saves.
"We tried to battle back, but at the end of the night, a lot of penalty calls against us that hurt momentum," New Jersey coach Lindy Ruff said.