Verhaeghe scored 4:27 into overtime with a wrist shot from inside the blue line, and the Panthers came back to defeat the Vegas Golden Knights 3-2 in Game 3 at FLA Live Arena on Thursday.
"To win it overtime in front of our fans, it gives us a little bit of momentum," Verhaeghe said.
Matthew Tkachuk had tied it 2-2 for the Panthers at 17:47 of the third period, scoring in front off the rebound of Verhaeghe's one-timer from the right point with Bobrovsky pulled for the extra attacker.
It's the Panthers' first win in a Cup Final. They were 0-6, including getting swept in the 1996 Final by the Colorado Avalanche.
Vegas leads the best-of-7 series, 2-1. Game 4 is here Saturday (8 p.m. ET; TNT, TBS, truTV, CBC, SN, TVAS).
"Over the course of his career he's gotten the puck off his stick faster than somebody can react to it," Florida coach Paul Maurice said of Verhaeghe. "So, you get to overtime, nobody is beefing you for a better play, 'I'm wide open, why didn't you hit me?' Just shoot the [darn] puck."
Verhaeghe got the puck from Sam Bennett high in the offensive zone and scored with a shot that got through traffic and beat Adin Hill clean on the glove side.
Florida killed a Vegas power play that extended 1:48 into overtime before Verhaeghe scored. Gustav Forsling was called for tripping Chandler Stephenson with 12 seconds left in the third.
Brandon Montour scored, and Sergei Bobrovsky made 25 saves for the Panthers, who are 7-0 in overtime in the Stanley Cup Playoffs.
"Everybody will probably say how they were leading most of the game, which they were, but at the end of the day nobody cares how we got here," Tkachuk said. "It's a 2-1 series. We came into this game to just win one game and we did that."
Jonathan Marchessault and Mark Stone each had a goal and an assist, and Hill made 20 saves for the Golden Knights.
Florida won despite going 0-for-5 on the power play and giving up two goals on its penalty kill.
The Golden Knights had opportunities in the third to extend the lead before Tkachuk scored. Bobrovsky made a save on a point-blank chance by Michael Amadio at 1:23, and Ivan Barbashev hit the crossbar on 2-on-1 with 4:18 remaining.
Vegas also had the power play in overtime and did not get a shot on goal.
"I thought we were doing a lot of things correct," Golden Knights coach Bruce Cassidy said. "We just couldn't get the next one to put the nail in the coffin, so to speak."