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.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut