Turris, acquired from Phoenix during the season, took a drop pass from Jim O'Brien and snapped a wrist shot past Henrik Lundqvist's glove from the left faceoff circle.
"We've got so much character in this room and that led us through," Turris said. "We rallied and tried to put pressure on them and chip away at their lead. It's exciting. It's a confidence builder and something to move forward on."
Milan Michalek and Sergei Gonchar scored in the second period for the Senators to erase a 2-0 deficit, and Craig Anderson stopped 31 shots.
"It's frustrating to lose in overtime, but there were some good things," Rangers coach John Tortorella said. "I think we still need some more guys involved, but it was two overtimes, the series is 2-2 and we're going back home."
Anton Stralman and Ryan Callahan had power-plays goals early in the first period for the Rangers.
"They had a good second period and we made it a little tougher on ourselves taking all the penalties," said Lundqvist, who made 28 saves. "They got some confidence on the power play and were moving the puck. They felt good about themselves and building their game from there. We regrouped and played really well in the third and did some good things but it wasn't enough."
The Senators also won Game 2 in overtime on Chris Neil's goal.
"We've had good starts and they're a good team, but we just need to find a way to win in overtime. That's what it comes down to," Rangers defenseman Marc Staal said. "We took a string of penalties in the second and that hurt our momentum a little bit. They've got a good power play and we've got to stay out of the box. We let them back in, but I thought we rebounded and had a good third, but we just couldn't find a way in overtime."
Both teams had great opportunities to take the lead in the third period. Ottawa created a flurry of chances in the final minute, but was unable to beat Lundqvist. New York's best opportunity came with just over 5 minutes remaining when Nick Foligno was called for tripping, but the Senators killed the penalty.
Anderson made big saves on Brian Boyle and Brad Richards early in the period, and Lundqvist made a huge save on Chris Phillips with just over 7 minutes remaining.
Ottawa cut it to 2-1 at the 7:04 of the second. Matt Carkner, coming out of the penalty box, took a pass from Jason Spezza and fed it to Michalek, who backhanded it over Lundqvist's right shoulder.
The crowd of 20,340 at Scotiabank Place gave Ottawa a standing ovation for more than 3 minutes.
With Ottawa on the power play late in the period, Staal caught Spezza in the face with a stick and then gave him an elbow to the head and no penalty was called. Spezza was hunched over when the hit took place, and he went down hard and left the game. He returned for the start of the third.
Ottawa tied it on the same power play, with Gonchar's shot from the top of the faceoff circle making it through traffic and past Lundqvist with 2:10 left in the period.
After going 1 for 11 on the power play in the first three games, the Rangers scored twice with the man advantage in the first period. Stralman opened the scoring at 49 seconds, and Callahan was left all alone at the side of the net to bang in a rebound at 6:10.
"Any time you have a lead, you don't want to lose it. But it doesn't matter right now," said Richards, who assisted on both New York goals. "It's over and it's time to move on to the next one. We'll take a look at it and build on the stuff we did well and see where we need to improve."
Notes: Canadian Prime Minister Stephen Harper was in attendance. ... Ottawa played the third period and overtime without center Jesse Winchester after he sustained an upper-body injury.
Associated Press
