Valtteri Filppula and Tom Pyatt also scored for the Lightning, who moved into a second-place tie with Montreal -- with one game at hand -- in the Atlantic Division after completing a 3-1-2 homestand.
Ben Bishop extended his single-season team record with his 32nd win.
"Great homestand, especially after you drop the first one," Tampa Bay coach Jon Cooper said. "It's not how you get there, you just need the points."
Vancouver, 10th in the Western Conference, is 3-6-1 in its last 10 games. Alexandre Burrows scored two goals, and Jannik Hansen added a short-handed tally for the Canucks.
"We know we still have a chance, but obviously we need to string wins together, and not just three, four or five," Hansen said. "We need to start winning games and a lot of them."
Burrows had an assist on Hansen's goal for a three-point game. He had a goal and two assists in Sunday's 4-3 shootout win at Florida.
After Pyatt scored midway through the third period, the Canucks got within one on goals by Burrows at 11:44 and Hansen at 15:16.
"We've still got to fix some things," Stamkos said. "You take a lead into the third we know, especially teams that are fighting for their playoff lives, they're going to throw everything at us. We have to be a little smarter."
Stamkos put the Lightning up 1-0 with a power-play goal from the low slot 17:56 into the game. Palat, who assisted on the goal, has five assists and nine points during a five-game point streak.
After Eddie Lack stopped eight shots early in the second, Tampa Bay went ahead 2-0 when Palat's pass went off sliding Canucks defenseman Ryan Stanton and into the net at 6:54.
Filppula made it 3-0 with 9 seconds left in the second when his pass deflected off Stanton and past Lack. The center has two goals and eight points during a five-game point streak.
"The first one, he's in the middle of the ice, taking ice away," Canucks coach John Tortorella said. "If he's going to go down and try to take away that in the short ice in a two-on-one, he's got to be more off to the post so it can't happen that way.
"The second one that goes in, it's just a tough break. A little bit of puck luck there, but it doesn't matter. We didn't get points, and we need points."
Burrows ended Bishop's bid for a second consecutive shutout on a power-play rebound goal 5:44 into the third. Bishop got his fifth shutout this season in a 3-0 victory over New Jersey on Saturday.
"You look at the checks and balances at the end of the day, it is who won and who lost. And we won," Cooper said.
The Lightning held a pregame ceremony honoring the 2004 Stanley Cup championship team. Tortorella coached Tampa Bay in 2004 and walked to center ice to a loud ovation when he was introduced. He had dinner with his former players Sunday night. "I'm not a big reminiscent-type guy, but when I sat with the guys last night and had that opportunity, it was truly a nice moment for me to be with them," Tortorella said. ... Stamkos has two goals in six games since he returned from a broken right shin. ... Filppula's goal was his 300th NHL point.
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