Tyler Johnson and Nikita Kucherov also scored to help the Lightning improve to 7-1-2 in the last 10 games.
Roberto Luongo stopped all three Carolina skaters in the shootout after making 23 saves.
The Hurricanes have lost six of seven to fall into last place in the Metropolitan Division.
Jordan Staal, Elias Lindholm and Viktor Stalberg scored for the Hurricanes.
Jeff Skinner scored for the last-place Hurricanes, who have their second five-game losing streak in the past five weeks.
The Stars ended a four-game losing streak, their longest this season. Carolina has lost two in a row.
The Hurricanes game against the Boston Bruins will still be played as scheduled Sunday evening.
“It’s been a joke around the locker room that I might go in today. And for it to actually happen, I kind of pinched myself.”
The Hurricanes scored six goals in the third – including four in a 4:40 span – to erase a 5-2 deficit.
The Hurricanes ran their home winning streak to five, and ended a six-game skid against Florida.