Nick Spaling and Bobby Butler scored for the Predators, who have lost a franchise-high seven consecutive games. They have won just once in their past 11 games.
Monday's loss eliminated the Predators from postseason contention. Nashville had qualified for the playoffs in seven of the past eight seasons.
"We still have to play hard," Nashville captain Shea Weber said. "It's obviously tough. We have to find a way to battle here in the last games and try to get some wins."
Prior to the singing of the national anthems, a moment of silence was observed in remembrance of the victims of Monday's bombings at the Boston Marathon.
Kesler gave the Canucks a quick lead at 1:34 of the opening period.
With the Canucks on a power play, Henrik Sedin slid a pass from the top of the left circle to Kesler in the high slot. From there, Kesler beat Nashville goaltender Pekka Rinne to the stick side with a wrist shot through traffic.
"I just shot through a screen there," Kesler said. "Great play by Hank and Danny to get me the puck and I found the back of the net."
Roy made it 2-0 for Vancouver lead three minutes later.
Former Predators defenseman Dan Hamhuis' shot from the high slot was kicked aside by Rinne, but Roy was able to dive and backhand the puck by Rinne just inside the right post.
The assist extended Hamhuis' point-scoring streak to six games, a career high for the defenseman Nashville selected with the 12th pick in the 2001 draft.
Spaling cut the Predators' deficit to 2-1 at 1:34 of the second.
Sergei Kostitsyn had the puck behind the Vancouver net. His centering pass found Spaling alone in front, where he one-timed the puck by Luongo for his eighth goal of the season.
Butler tied the game at 2 at 11:40 of the second.
Carrying the puck down the right side, Butler cut toward the faceoff circle where he fired a wrist shot by Luongo on the far side.
"We got off to a slow start and dug ourselves a hole and we found a way to fight back," Nashville coach Barry Trotz said. "In the second period we responded really well, and you could see right to the end."
Vancouver regained the lead on Garrison's power-play goal at 17:02 of the second.
"We were good in the first, and then for whatever reason, our play slipped in the second until Henrik's line went out there and had a real strong shift," Vancouver coach Alain Vigneault said. "That enabled us to go on the power play and score a goal."
Burrows gave Vancouver a 4-2 lead 50 seconds into the third when he redirected Daniel Sedin's shot by Rinne.
"We have done that before when we were high and really don't have a great shot on net," Burrows said. "We were trying to get a wrist shot for a tip and I was able to get a stick on it and luckily it went in."
Kesler scored his second of the game at 15:53 of the third with a wrist shot from the right faceoff dot.
Nashville D Kevin Klein missed his first game of the season with what Predators coach Barry Trotz called a "below the neck injury" at the morning skate. Mattias Ekholm, who played his first game with the Predators this season, replaced him in the lineup. ... The Predators are 0-12-1 when trailing after the first period this season. ... The Canucks have not allowed a power-play goal in their past six games.
Copyright by STATS LLC and The Associated Press
NASHVILLE, TN - APRIL 15: Head coach Barry Trotz of the Nashville Predators waits with the team before taking the...