The Penguins, who have allowed at least four goals in five of their past six games, went 0-for-5 on the power play and gave up a shorthanded goal.
"We knew the game we were up against," Penguins coach Mike Sullivan said. "Every team in this league plays hard. Give [the Devils] credit; they played hard tonight, they defended hard. I think we have to do a better job in those areas of our game so that we can bring more consistency. When we talk about forging an identity, those are the very things that we talk about.
"For me, the foundation of the game is just rooted in competitive spirit. You have to be willing to invest emotionally first and foremost. When you do, that's usually when players and teams are at their best."
Zajac gave the Devils a 1-0 lead at 13:08 of the first period on a wrist shot from the left face-off circle.