Jaromir Jagr gave a goal and three assists while Ryane Clowe had a goal and two assists for the Devils, who had dropped three of four but now have 79 points and are three behind the Columbus Blue Jackets for the final wild card spot in the Eastern Conference.
Martin Brodeur started in net, but was pulled in the second period after giving up three goals on nine shots. He did get the win, though, thanks to Cory Schneider stopping all 13 shots he faced in relief.
It took just 12 seconds for the Devils to take the lead, as a right circle faceoff win saw the puck go back to Mark Fayne, who fired a shot that Jagr tipped on net, with Zajac jamming home the rebound from the left side.
The Devils made it a 2-0 game at 9:24 of the first when Jagr's snap shot from the right circle found the back of the net, and Zajac netted his second of the game two minutes later on the power play with a one-timer from the left circle.
Florida got on the board with 2:38 to play in the first, as Kulikov's wrister from the right circle found the left corner of the net for a shorthanded marker.
The Panthers made it a 3-2 game just 50 seconds into the second period, as Boyes fought through one defender at the right boards, stickhandled past another before snapping home a backhand for his 20th of the season.
New Jersey got the goal back 2:15 later after a turnover in the neutral zone saw Jagr push the puck forward for Zajac, who passed it from the right circle to the left where Clowe wristed it in.
Florida, though, answered 3:30 later as a shot from the left point by Dylan Olsen was re-directed in by Pirri.
That prompted a goaltending change for New Jersey, as Brodeur exited the net and Schneider took over.
New Jersey notched a shorthanded goal of its own at 4:02 of the third, as Jacob Josefson's one-timer from the slot found the back of the net for his first goal of the season.
Zajac capped the hat trick near the halfway mark of the third, when he followed up his own rebound by putting the puck in from the left side for his 16th of the season.