Montour received a pass from Shane Wright in the left face-off circle and buried a low wrist shot through the pads of Andrei Vasilevskiy for the game-winner.
Bobby McMann and Kaapo Kakko each had a goal and an assist for the Kraken (32-29-10), who ended a four-game losing streak (0-3-1). Philipp Grubauer made 30 saves.
Jake Guentzel had a goal and an assist for the Lightning (44-21-6), who are 4-0-2 over their past six games. Charle-Edouard D'Astous had two assists, and Vasilevskiy made 15 saves.
Montour gave the Kraken a 1-0 lead at 15:18 of the first period. Frederick Gaudreau skated into the offensive zone and fed a pass to Montour, who snuck behind the defense by cutting through the right circle and beat Vasilevskiy with a backhander to the blocker side.
Anthony Cirelli tied it 1-1 at 17:01. Brandon Hagel threw the puck to the net and it deflected off the stick of Kraken defenseman Ryan Lindgren and rolled to Cirelli, who chipped it past Grubauer's blocker from in front.
Kakko put Seattle up 2-1 just 1:11 later at 18:12, burying a centering pass from Chandler Stephenson past a sprawling Vasilevskiy from the slot.
McMann made it 3-1 with his sixth goal in seven games at 3:35 of the second period. Adam Larsson's point shot hit off Kakko and McMann collected the rebound and stuffed it in at the right post behind Vasilevskiy.
Guentzel cut the deficit to 3-2 at 5:56, finishing a stretch pass from Darren Raddysh on a breakaway by snapping the puck over Grubauer's glove.
Corey Perry tied the game 3-3 with a power-play goal at 10:18 when he beat Grubauer with a snap shot to the glove side that went under the bar from the slot.
