Steve Bernier scored for the Devils, and Martin Brodeur made 18 saves, but New Jersey has lost two straight for the first time since a three-game skid from Jan. 19-24.
"It's a missed opportunity," Devils coach Peter DeBoer said. "We didn't do enough, made a couple mistakes, and they made us pay."
Bernier and Bergenheim traded goals in the first period.
Bernier scored his first goal since Feb. 10, 2011, 1:39 in to give the Devils a quick lead. Dainius Zubrus forced a turnover by Keaton Ellerby below the right circle, and Zubrus' pass to Alexei Ponikarovsky deflected in off Bernier in front.
New Jersey surged after the early goal, and the Devils had an 8-0 advantage in shots until Brodeur stopped Erik Gudbranson's drive at 10:58 of the first. But while the Devils dominated, they couldn't solve Clemmensen. The Panthers' play improved as the first period progressed.
"That's what you need for your goaltender," Panthers defenseman Brian Campbell said. "It was an important game for us. It is important to get wins."
The Devils recorded only 15 shots in the final 40 minutes.
"I think we came out well in the first 10 minutes," DeBoer said. "I think it was a combination of a couple things. We're playing our seventh game in 11½ days (and) I think we ran out of gas. On the other side, I thought Florida did a good job of bottling us up. I think it was a combination of those two things.
"We just couldn't generate."
Bergenheim got the Panthers even at 15:08 with a power-play goal, redirecting Gudbranson's shot past Brodeur.
Versteeg gave the Panthers a 2-1 lead 7:43 into the second when he ripped Stephen Weiss' blocked shot over Brodeur from the slot. Versteeg's goal was his 21st of the season, one shy of his career high set in the 2008-09 season when he was with Chicago.
"I think Steeger is a big-game player, a big-game goal scorer. He played on both sides of special teams," Dineen said. "I was happy with the way he responded."
Florida protected its one-goal lead in the third period by clogging the neutral zone and keeping the Devils' shots from the perimeter.
"We did a good job coming back," Campbell said, "blocking shots, doing what (good teams) do."
Samuelsson added an empty-net goal at 19:35 to make it 3-1.
"A game like this late in the year is better than a game like this earlier in the year," Clemmensen added. "For us to do it here is good. It's a road win; good for us."
Florida was 1 for 3 on the power play and killed New Jersey's only advantage.
The matinee was Florida's first of the season. The Panthers will play another day game on Sunday at the New York Islanders. ... The teams split the season series 2-2. ... Florida D Jason Garrison sustained an undisclosed lower body injury in the second period and didn't return.