Paul Martin scored the other goal while Scott Clemmensen made 23 saves for the Devils, who have won two straight on the heels of a four-game losing streak.
"We found a way to win," said New Jersey head coach Brent Sutter. "We had a very good first period. We kind of lost it a little bit in the second and got it back in the third. It was one of those games where we had to scratch and claw."
"I think we had a bad first period and the coach was really upset," said Frolik. "In the second period it was much better, we got more shots, more hits. I think if we play like that every period we can win the game. But it didn't happen, we have to forget it and be ready for tomorrow."
Holding a 1-0 lead early in the third period, New Jersey solidified its advantage with a pair of goals separated by 62 seconds.
Martin made it a 2-0 contest as he faked a shot from the right point to get around a defender and from the slot snapped a wrister past Vokoun at the 5:49 mark.
A bit later, Florida turned the puck over in its own end. Gionta wound up with the puck and passed it over to the left side for Elias, who fed it back to Gionta in the slot for the easy tap-in.
Florida got on the board thanks to the power play about two minutes later as Frolik blasted a slap shot from the right point through a screen and past Clemmensen for his first goal.
Neither side got on the scoreboard in the first period with Vokoun making 12 saves and Clemmensen stopping five shots.
New Jersey grabbed the lead at the 6:47 mark of the second period. Good cycling work resulted in Gionta with the puck along the right boards. Gionta threw the disc out front to Elias, who shifted to the backhand and slipped the puck past Vokoun for his eighth goal of the year.
Clemmensen had a strong second period as he made 10 saves to keep the Panthers off the board.