Joe Pavelski also scored twice and helped out on one of Couture's markers as the Sharks broke the Western Conference quarterfinal game open with three quick goals early in the third period.
Patrick Marleau added a score and Joe Thornton notched two assists for the sixth-seeded Sharks, who pushed the Northwest Division champion Canucks to the brink of post-season elimination with the victory.
Antti Niemi stopped 28-of-30 shots for San Jose, which will look to sweep the series at home in Tuesday's Game 4.
Canucks goalie Cory Schneider was pulled in the third period after yielding all five goals on 28 shots.
Roberto Luongo made 10 saves the rest of the way.