After a scoreless overtime, Filppula went first in the tiebreaker and bested Canadiens goaltender Carey Price, putting a shot into the left side of the net.
Ben Bishop, who finished with 28 saves, then made stops on Danny Briere and Brendan Gallagher as the Bolts responded to the loss of their top goal-scorer with a fifth victory in six games.
Stamkos, the NHL's co-leader in goals and assists, underwent surgery on a broken leg Tuesday and is out indefinitely.
Ryan Malone scored in the first period for Tampa Bay.
Briere scored late in regulation and Price made 44 shots for struggling Montreal, which fell for the fifth time in six games.