Martinook put Carolina up 3-2 when he threw the puck on net from a sharp angle in the left face-off circle and it deflected in off Tampa Bay defenseman Victor Hedman.
Sebastian Aho and Andrei Svechnikov each had a goal and an assist for Carolina (42-18-6), who have won four of their past six. Frederik Andersen made 17 saves.
Yanni Gourde and Charle-Edouard D'Astous scored for the Lightning (40-21-4), who have lost seven of their past nine games. Vasilevskiy made 31 saves.
Svechnikov put the Hurricanes up 1-0 just 36 seconds into the first period during a delayed penalty. He snapped Aho's centering pass from the right corner to the short side after Hedman impeded him on a breakaway.
Aho made it 2-0 at 17:28. K'Andre Miller sprung Aho on a breakaway with a stretch pass from the defensive zone, but he was stopped by Vasilevskiy. Svechnikov gathered the rebound and got the puck back to Aho, who scored into an open net from the right of the crease.
The Lightning scored two goals in 1:21 span to tie the game.
Gourde cut the deficit to 2-1 at 14:10 of the second period with a wrist shot from the right face-off circle under the right arm of Andersen.
D'Astous evened it 2-2, finishing a 2-on-1 rush when he tapped in Brayden Point's centering pass off Andersen's blocker as he crashed the net.
Logan Stankoven secured the 4-2 final with an empty-net goal at 18:29 of the third period after Andersen made a point-blank save on Nikita Kucherov's shot from in tight.
