Campbell scored from the inside edge of the left circle as the trailer on a 3-on-2 rush. It was his second goal of the season.
Goaltender Roberto Luongo made 28 saves for Florida (10-9-4).
"We felt we had played well," Luongo said. "Just in the third period, we wanted to keep doing what we had been doing. Their goalie was playing pretty well, so we wanted to get some bodies in front of the net. ... It feels good to be able to tie it up in the third period, make a few saves and be rewarded with the win in overtime."
Dylan Larkin scored for Detroit (12-8-4), which has played four straight overtime games and six in the past seven. Mrazek made 30 saves.
"I didn't think we were good enough tonight," Red Wings coach Jeff Blashill said. "I think we can play much better than we played. I think the second half of the first period was where we played the best. We had a lot of zone time during that stretch. It seemed like we just didn't have enough energy, we didn't win enough puck races, didn't win enough puck battles. So overall, we still sat with a lead late and disappointed to give up that lead. But we need to be better for 60 [minutes] before that."
Red Wings forward Gustav Nyquist nearly broke the tie when he hit the goal post with 2:28 left in the third period.
Larkin gave Detroit a 1-0 lead at 7:02 of the second. He intercepted Aleksander Barkov's pass up the middle about 10 feet inside the Florida blue line, skated in and beat Luongo to the stick side. It was Larkin's 10th goal, which leads NHL rookies. He has six goals and one assist in his past seven games.
The Red Wings have surrendered a third-period lead in five of their past seven games.
"I think everyone is frustrated with the loss," Larkin said. "I think it seems like it's happening more and more and I think we just have to bear down and close out the game."
The Panthers outshot the Red Wings 22-18 in the second and third periods, but Mrazek was solid. He helped kill off a 1:27 5-on-3 Florida power play in the second period, then denied Smith from the left circle and Brandon Pirri on the rebound 4:14 into the third.
"We have to close these games early in the game rather than later," Mrazek said. "You cannot wait until the end of the game. It happened a couple of time this year that we let some goals in.”
Smith thought he scored 28 seconds into the game, but the play was ruled no goal because Panthers forward Jonathan Huberdeau interfered with Mrazek. Gallant challenged the call, but it was upheld.
"When I looked at it, it was a 50/50 call," Gallant said. "(It was) early in the game and (I) wanted to create some momentum."
|Power Play %||20.5%||15.8%|
|% on Road||8.6%||21.2%|
|% at Home||29.2%||11.6%|
|Mike Green Holding against Reilly Smith|
|Jonathan Huberdeau Interference against Mike Green|
|Mike Green Embellishment against Jonathan Huberdeau|
|Aleksander Barkov Holding against Henrik Zetterberg|
|Tomas Jurco Tripping against Brian Campbell|
|Mike Green Hooking against Jonathan Huberdeau|
|Darren Helm Tripping against Vincent Trocheck|
|Brian Campbell Tripping against Tomas Tatar|
|Jussi Jokinen Unsportsmanlike conduct against Tomas Tatar|
|Tomas Tatar Unsportsmanlike conduct against Jussi Jokinen|
|Justin Abdelkader Roughing against Erik Gudbranson|
|Erik Gudbranson Roughing against Justin Abdelkader|
|Darren Helm Hi-sticking against Reilly Smith|
|Logan Shaw Tripping against Pavel Datsyuk|
|Nick Bjugstad Hooking against Justin Abdelkader|