After a scoreless overtime, the game went to a shootout.
Couture bested Corey Crawford in the first round and Pavelski notched the game-winner in the third as the Sharks ended a string of six straight losses at United Center.
Alex Stalock turned aside 33-of-35 shots and made a save on Jonathan Toews in the tiebreaker to earn the win.
Jason Demers also scored in regulation for San Jose.
Crawford finished with 40 saves.
Boston schools pull out free condoms over wrapping complaints
Millions of Getty images now available for free via embed tool