Ondrej Palat and Steven Stamkos each had a goal and an assist, Ben Bishop stopped 25 shots and the Lightning finished a six-game stretch at home with three consecutive wins and eight of a possible 12 points.
The win propelled the Lightning (37-25-7) into second place in the Atlantic Division. Tampa Bay and the Montreal Canadiens each have 81 points, one more than the Toronto Maple Leafs, but the Lightning have a game in hand on each.
"We were a little loose in the third, but we've got points in five consecutive games at a time when we really need them so hopefully this is some good momentum heading on the road," Lightning coach Jon Cooper said. "We got to take advantage of a team playing their fourth game in six nights, all over the country. You feel for them a little bit, but not too much because we go through the same thing. What comes around goes around. You have to give them credit; they threw everything at us in the third."
Vancouver had the first eight shots in the third period and produced some quality scoring chances, including a breakaway attempt by Nicklas Jensen that Bishop denied at 3:42.
"It's a game of bounces and you have to deal with it," Burrows said, "but at this time of year, you wish the bounces would go your way."
But the barrage of shots finally paid off when Tom Pyatt was penalized for goaltender interference and the Canucks connected on the power play at 5:44 to cut the margin to 3-1. Burrows flipped a rebound of Kevin Bieksa's shot from the blue line past Bishop for his fourth goal of the season.
Pyatt made amends just over three minutes later when his bad-angled shot beat Lack from the short side, restoring the Lightning's three-goal lead. It was Pyatt's third goal of the season.
"Obviously, I want the third goal back," Lack said. "I thought [Pyatt] was going to pass it, but he fired it quickly. If I would have had that, it would have been a 3-3 tie."
The Canucks (31-30-10) cut the deficit back to two with Burrows' second goal of the game at 11:44 and then added a shorthanded goal by Jannik Hansen, his 11th of the season, at 15:16.
"We fought back and came up empty," Vancouver coach John Tortorella said. "We can't come up empty."
Vancouver's third-period rally concerned Stamkos. He was clear that there is some work to be done before Tampa Bay visits Toronto on Wednesday.
"We still have to fix some things," the Lightning captain said. "We take the lead into the third. We know especially teams that are fighting for their playoff lives are going to throw everything at us. We have to be a little bit smarter."
Tampa Bay scored twice in the second period to extend its lead to 3-0.
After Bieksa fanned on a 2-on-1 chance, the Lightning turned the tables and Palat finished a 2-on-1 break the other way when he deflected Matt Carle's feed off Vancouver defenseman Ryan Stanton and past Lack at 6:54 while the teams skated 4-on-4.
Stanton was victimized again in similar circumstances late in the period. Bishop made a stellar stop on Zack Kassian and the Lightning took off in the other direction. This time, Valtteri Filppula was the beneficiary. His centering pass hit Stanton's skate and found its way past Lack for Filppula's 22nd goal of the season at 19:51.
"First one, he's in the middle of the ice, taking the ice away," Tortorella said of Stanton. "If he's going to go down and try to take away the ice in a short 2-on-1, he has to be more off to the post so it can't happen that way. The second one? That was just a tough break. A little bit of puck-luck there. Doesn't matter what it is, we didn't get points and we need points."
The first period played out evenly until Kassian was called for tripping. On the ensuing power play, the Canucks left Stamkos alone to the right of their net and when the puck bounced his way, he didn't miss. Lack had no chance on Stamkos' 16th goal of the season at 17:56. With an assist, Palat extended his scoring streak to five games; his goal in the second period gave him eight points over that stretch (three goals, five assists).
Vancouver defenseman Christopher Tanev was injured in the opening period and did not return.