Game Two is set for Tuesday, also on Panthers home ice.
Verhaeghe scored on the power play off a pass from Aleksander Barkov, who had two assists. It increased Verhaehge's team record postseason goal count to 16.
The Panthers also got goals from Sam Reinhart and Matthew Tkachuk, who added an assist. Sergei Bobrovsky made 17 saves.
Reinhart was second in the NHL with 57 regular-season goals. Florida was also first in the league in lowest goals-against average this season (2.41).
Tampa Bay, which has never lost a series to the rival Panthers - winning in 2021 and again in 2022 - got goals from Brandon Hagel and Steven Stamkos.
Andrei Vasilevskiy made 25 saves for the Lightning.
Florida opened the scoring 6:17 into the first period as Gustav Forsling's shot was deflected in by Reinhart.
Tampa Bay, after just one shot on net in the first 16 minutes, posted three shots in three seconds, tying the score 1-1.
The sequence started when Anthony Cirelli won a faceoff from the right side. Hagel tapped the puck back to Stamkos for the first shot. Cirelli's rebound attempt was stopped, but Hagel got the loose puck and beat Bobrovsky stick-side.
Florida took a 2-1 lead just 58 seconds into the third period on a power play that carried over from the second. Prior to that goal, the Panthers had gone just 4-for-53 on their power play in the last 16 regular-season games and their first two periods on Sunday.
On the go-ahead score, Verhaeghe looped all the way around and snuck in back door for a tap-in off of Barkov's perfect pass. Barkov's feed got past Tampa Bay's Emil Lilleberg and Matt Dumba.
Tkachuk's empty-net goal with 2:05 left gave Florida a 3-1 lead.
After a penalty on Verhaeghe for cross-checking Nikita Kucherov, the Lightning got a power-play goal from Stamkos.
But since the goal came with just 9.3 seconds left, Tampa Bay ran out of time.
