Three unanswered goals by the Sharks gave them a 3-1 advantage as time was running down in the second period but Troy Brouwer and Brian Campbell scored for Chicago in the final 23 seconds of the stanza to bring about a 3-3 tie.
Thornton broke the deadlock by tipping in a shot 7:13 into the third and Clowe clinched it with an empty-net goal in the closing minute.
Antti Niemi, who helped bring Chicago the Stanley Cup last season, made 26 saves to get the victory. It was his third win this season against his former team.
The Blackhawks have lost two straight after winning four in a row. Marty Turco stopped 20 shots in the loss.