Joe Pavelski added a goal and an assist for San Jose (43-28-6), which clinched a playoff spot a night earlier with 5-2 home victory over the Los Angeles Kings. Brent Burns chipped in with two assists.
Chris Higgins replied for Vancouver (27-36-13), which is 0-8-1 during its current slide.
Jacob Markstrom stopped 27 shots in the loss.
The Sharks, who host the Canucks on Thursday, sit third in the Pacific Division, two points back of the Anaheim Ducks and three adrift of L.A.
Limping towards the finish line of a lost season that will see them miss the playoffs for the second time in three years, the Canucks sit 28th in the overall standings, tied for the fewest points (67) with the Toronto Maple Leafs and Edmonton Oilers.
After a scoreless first period where the Canucks had a number of chances to grab the lead, Pavelski got San Jose on the board 20 seconds into the second. Burns' point shot hit Markstrom's glove and bounced off Pavelski, who swept home his 37th of the season.
Recalled from the AHL earlier this month after getting demoted in January, Higgins scored his third of the season and first since Nov. 8 when he ripped a shot over Reimer's glove for a short-handed goal at 6:21.
The goal came with Vancouver down a man after rookie forward Jake Virtanen was assessed a five-minute interference penalty and a game misconduct for a blindside hit on Sharks defenceman Roman Polak.
Couture restored San Jose's lead with 11 seconds left in the period when he caused a turnover behind the net and slid his 11th of the season past Markstrom after a nice kick pass from Nick Spaling in front.
The Canucks had a power play early in the third period, but Reimer made a couple of nice stops to keep the visitors in front before Vancouver rookie Jared McCann sent a backhand over the net on great chance a few minutes later.
Couture then beat Markstrom between the legs on a 5-on-3 man advantage with 5:59 left in the third to seal it with his 12th.
He completed the hat trick into an empty net with 16.4 seconds remaining on the clock.
Without a number of injured regulars -- including defencemen Alexander Edler and Luca Sbisa, along with forwards Brandon Sutter, Radim Vrbata and Derek Dorsett -- the Canucks have been forced to go with their youngsters as they play out the string.
Vancouver had a bright start against San Jose and a couple of chances to grab a lead in the first after Joonas Donskoi hit the post for the Sharks less than a minute in.
Daniel Sedin couldn't control a bouncing puck at the side of the net, Alexandre Burrows was robbed by Reimer's glove and Bo Horvat made a nice move before firing a shot that just grazed the San Jose goalie's shoulder and stayed out.
The Sharks had the period's final chance when Couture jumped on a turnover, but Markstrom was there to close the door.