San Jose scored five times on 13 shots during the opening 20 minutes and quickly drove Tampa Bay goaltender Mathieu Garon from the game. Garon surrendered three of the first-period goals on eight shots.
The Sharks won their third in a row and halted Tampa Bay's two-game winning streak.
San Jose's five-goal outburst came in the span of 11:11 and was started by Benn Ferriero 3:26 into the contest.
Joe Pavelski then scored on the power play and Andrew Desjardins made it 3-0. Couture followed with his 13th and 14th goals of the season before Domonic Moore tallied for the Lightning.
Antti Niemi got the easy victory by making 25 saves. Dwyane Roloson came on to replace Garon and stopped 30-of-34 shots.