Turris' shot from the sideboards sneaked past Montreal backup goalie Peter Budaj, who came on for the injured Carey Price at the start of overtime.
Cory Conacher scored with 22.6 seconds to go in regulation to force overtime. Mika Zibanejad had the other goal for the Senators, who got 26 saves from Craig Anderson.
P.K. Subban and Alex Galchenyuk scored 62 seconds apart in the second period for Montreal. Price made 30 saves two nights after allowing all six goals in Ottawa's 6-1 victory in Game 3. He was injured on Conacher's tying goal late in the third period.
Game 5 goes Thursday night in Montreal in a series that has seen a little bit of everything.
Down 2-0 in the third period and with little going their way, the Senators finally beat Price when Zibanejad's goal at 11:55 off a pass from Chris Neil. The puck went in off Zibanejad's skate, but was confirmed after video review.
Ottawa continued to push and tied it in a wild scramble with Anderson on the bench, sending the crowd of 20,500 at Scotiabank Place into a frenzy.
After carrying the play for long stretches of the first period, the Canadiens were finally rewarded in the second. Subban, who took 25 minutes in penalties in Sunday's fight-filled affair and was booed every time he touched the puck, opened the scoring, taking a feed from Tomas Plekanec off the rush and roofing a shot past Anderson.
The flashy defenseman celebrated by mimicking a high-five to a Canadiens fan seated by the glass before being mobbed by his teammates.
Galchenyuk then made it 2-0, again beating Anderson up high, much to the delight of the large contingent of Montreal fans in attendance.
But they had little to cheer the rest of the way.
Anderson made a big save to keep the deficit at two a few minutes later on defenseman Andrei Markov.
Senators defenseman Eric Gryba returned to the lineup after serving a two-game suspension for his hit on Lars Eller. Gryba caught Eller with his head down at the Montreal blue line just as he received a pass in Game 1, sending the Canadiens forward flying.
The Canadiens said they wouldn't be looking for retribution on Gryba, but Montreal forward Brandon Prust took an undisciplined roughing penalty late in the period, only to see the Senators' power play fail to capitalize.
Notes: The Canadiens were missing forwards Brian Gionta and Ryan White with upper-body injuries. Jeff Halpern and Gabriel Dumont drew into the lineup in their place. ... Gryba took the spot of Senators defenseman Patrick Wiercioch, who left Game 3 with a lower-body injury. There were no other lineup changes for Ottawa. ... Eller, who was sent to hospital with a concussion, broken nose and some lost teeth after the Gryba hit in Game 1, remains out. ... Keila Penner and her family were the guests of Senators owner Eugene Melnyk at the game. The 11-year-old was sent home from her Montreal-area school last week for refusing to remove her Ottawa sweater on "Habs Jersey Day." Keila's father grew up in Ottawa and is a diehard Senators fan.
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