The Bolts struck first on a power play goal from Brayden Point. A puck in the slot found its way to Point's stick and he quickly fired a shot past Andersen before the goaltender could recover.
Nazem Kadri equalized for the Leafs two minutes later. Andrei Vasilevskiy couldn't control a rebound and Kadri pounced on the puck, scoring on a backhand.
The Bolts and Leafs went back and forth trading goals in the second period.
Patrick Marleau gave Toronto the lead 13 minutes into the second period. With traffic stationed out in front, Marleau fired a shot that Vasilevskiy couldn't pick up.
It only took 37 seconds for the Lightning to answer back, however. Victor Hedman took a drop pass from Nikita Kucherov and beat Andersen to the blocker side to tie the game back at two.
Mitch Marner then scored the eventual game-winning goal just over a minute after Hedman tied the game. Marner was able to station himself unmarked in front of Vasilevskiy and redirect pass from John Tavares past Vasilevskiy to give Toronto the lead for good.
Down a goal to start the third period, the Lightning generated plenty of scoring chances but Andersen stopped all 16 shots they sent his way in the final frame. Most of the pressure came late in the game with Vasilevskiy out of the net but Tampa Bay ultimately couldn't break through.
Zach Hyman added an empty net goal with 51 seconds to play to round out the scoring.
The loss snapped Tampa Bay's nine-game home winning streak.
Tampa Bay now moves on to face the Sharks on Saturday night in the final game a 10-day layoff for the bye week and All-Star Game.
Scoring Summary
1st Period
TBL | 9:48 Brayden Point (30) - Victor Hedman, Steven Stamkos - PPG
TOR | 11:48 Nazem Kadri (10) - Kasperi Kapanen, Patrick Marleau
2nd Period
TOR | 13:06 Patrick Marleau (11) - Nazem Kadri, Nikita Zaitsev
TBL | 13:43 Victor Hedman (6) - Nikita Kucherov, Brayden Point
TOR | 14:49 Mitchell Marner (19) - John Tavares, Andreas Johnsson
3rd Period
TBL | 19:09 Zach Hyman (8) - Nikita Zaitsev, John Tavares - EN
