Wheeler scored on Boston's second attempt after losing control of the puck while trying to deke Clemmensen. Wheeler's shot from a sharp angle then hit the post before bouncing in off Clemmensen's ankle.
"The puck went where I wanted it to go, but I just kind of didn't have the handle I wanted to on it," Wheeler said. "It just kind of snuck off my stick there. Sometimes it's better to be lucky than good, right?"
Thomas helped keep the game tied late in the regulation with several big saves in the final minutes, including stopping defenseman Dennis Wideman's tap-in on a 2-on-1 break.
Clemmensen, making his seventh start of the season and his first since Dec. 4, stopped 40 shots for Florida.
"It was one of those duels where both goaltenders stood tall," Boston coach Claude Julien said.
Krejci almost completed the hat trick in overtime, but Clemmensen made a great pad save to stop his shot from close range.
Boston won for the first time in four shootouts, while Florida dropped to 2-1.
The game marked the beginning of a five-game road trip for Boston, which also will play at Tampa Bay, Atlanta, Buffalo and Toronto.
"Our goal is to have a really successful road trip and you can't do that without a good first game," Wheeler said. "I think we were a little sluggish in the first period, but that was to be expected coming off Christmas season. Everyone recovered well and it's great to get those two points."
Boston trailed 2-1 entering the third period before Krejci tied it at 10:35 off a nifty pass from Wheeler.
Krejci took Wheeler's pass while skating in front of Clemmensen and beat the goalie with a deke to his backhand.
"Our goalie was just as good as their guy, but I think the story of the game was missed opportunities," Panthers coach Pete DeBoer said.
"We had opportunities at different points in the game to extend the one-goal lead to two, some point-blank chances. And [Thomas] either made a save or we just missed. You let a team like that hang around, that's what happens."
Santorelli broke a 1-1 tie at 15:51 of the second period. His goal was allowed after a video review to determine whether his tip of Frolik's wrist shot from the point was caused by a high stick.
"I was pretty nervous," Santorelli said. "Luckily, it [was allowed]. We needed that, so it was good."
Booth opened the scoring 1:02 into the second when he grabbed a loose puck in the Boston zone, spun around and beat Thomas with a wrist shot from the slot.
Krejci tied it at 9:05 of the second period on a rebound off a slap shot from the point by defenseman Zdeno Chara.
"Their whole team, from the goalie out, played well," Clemmensen said. "I could say the same for us. It could have gone either way, especially getting to overtime and into the shootout."
Boston C Brad Marchand missed his third consecutive game because of a lower-body injury. ... Florida C Stephen Weiss and D Bryan McCabe each had five-game point streaks snapped. ... Florida had been the only team in the league without an overtime or shootout loss.