Zajac, Zach Parise, Dainius Zubrus and Patrik Elias provided the scoring for the Devils, who won for the eighth time in nine games.
They avenged the lone loss in that span, a 4-1 defeat in Pittsburgh on Nov. 29.
Matt Cooke scored for Pittsburgh, ending Clemmensen's shutout bid with 3:58 remaining. Evgeni Malkin, the NHL leading scorer, bulled his way past two defenders to flip the puck across the slot to Cooke for the tap in.
The Penguins' losing streak reached three as they were beaten in regulation in New Jersey for the first time in seven games (5-1-1).
Zajac made it 1-0 with 2:21 remaining in the second, starting the sequence with a shot that sailed wide left of the Penguins net.
Pittsburgh goalie Dany Sabourin couldn't control the carom off the end boards and was helpless as the puck came right back to Zajac for the goal.
It was another frustrating period for captain Sidney Crosby and the rest of the Penguins, who had their chances against Clemmensen.
Crosby had three excellent opportunities in the period: a close in power-play bid, a backhander after slipping behind defenseman Paul Martin and a wide-open shot that hit the post.
In the third, Parise deflected Bryce Salvador's shot from the left point past Sabourin at 2:18 for his team-leading 16th goal.
Zubrus tallied at 10:12 with Salvador picking up another assist.
Elias added an empty-net goal in the final seconds.
