Sturm's game-winner marked the 15th time Montreal has lost this season after having the lead or being tied in the third period.
In this one, the Canadiens took a 2-1 lead after two periods before San Jose got goals from Adam Graves and Sturm.
Sturm ended San Jose's two-game losing streak on a wraparound. After former Canadien Vincent Damphousse broke free from defenseman Francis Bouillon in the right corner, he skated behind the net, and left the puck for Sturm, who poked it through the pads of goaltender Jose Theodore.
Montreal had its winless streak reach eight games (0-6-1-1), fell to 0-2-0 on a pivotal seven-game road trip, and dropped seven points behind Boston and the New York Islanders for the final Eastern Conference playoff berth.
Bouillon's goal with 95 seconds left in regulation salvaged a point for the Canadiens. He also made a nice defensive play on Sturm before losing Damphousse in overtime.
Donald Audette and Chad Kilger also scored for Montreal, whose 88 third-period goals allowed lead the league.
Patrick Marleau had the other goal for San Jose, which won for just the third time in 26 games when trailing through two periods.