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.
Seattle is three points behind the Nashville Predators for the second wild card into the Stanley Cup Playoffs from the Western Conference.
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.
Tampa Bay remained in second in the Atlantic Division, two points behind the Buffalo Sabres and four points ahead of the Montreal Canadiens.
"Since we went on the road we haven't lost in regulation in our last six and played some pretty emotional games with a lot of travel in there," Lightning coach Jon Cooper said. "We need points to keep our head above water here and they keep getting them. You want to be greedy and get both."
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.
"We're a desperate team right now and we have to be engaged on every shift," Montour said. "We got two points, we move on."