The Senators were trailing 2-0 midway through the third period when they launched a late comeback.
Mika Zibanejad found the net with 8:05 remaining in regulation, halving the Sens' deficit, and Cory Conacher tallied the equalizer after a scramble in front the net with just 23 seconds remaining.
The Sens completed the rally in overtime on Turris' wrist shot past Peter Budaj, pushing the Canadiens to the brink of elimination in their Eastern Conference quarterfinal series.
Turris also scored an overtime game-winner for Ottawa in last year's first-round Game 4 triumph against the New York Rangers.
Craig Anderson turned aside 26-of-28 shots in the victory.
P.K. Subban and Alex Galchenyuk found the net for Montreal, which will attempt to stay alive at home in Thursday's Game 5.
Carey Price stopped 30-of-32 shots in regulation, but was replaced in net by Budaj in the overtime. The back-up netminder made one save before yielding the game-winner.