Alex Kovalev and Maxim Lapierre scored on Montreal's first two attempts in the tiebreaker after Jaroslav Halak turned aside Ottawa's first two shots to give the Canadiens a 5-4 win over the Senators on Saturday night.
Halak stopped Daniel Alfredsson, and Lapierre scored on a backhand as the Canadiens salvaged the win after blowing a 4-2 lead in the third.
"It's something that we've talked about. Going into the third period with the lead, we want to be able to shut the door and take pride in playing defensively," Canadiens defenseman Mike Komisarek said. "More than anything, it was a drop off in our game a little bit and they picked up their game, and it showed."
Halak made 23 saves in his eighth straight start in place of injured All-Star starter Carey Price.
"He's making the save when it needs to be done and he's winning games, so that's the most important thing," Canadiens coach Guy Carbonneau said.
"The biggest thing is we got the two points, and Boston lost tonight, so that's huge for us," Lapierre said. "We're on a hot streak right now. It doesn't matter how we get the points."
The Eastern Conference-leading Bruins, beaten 2-1 at Washington on Saturday night, are 10 points ahead of Montreal in the Northeast Division.
Mike Fisher scored Ottawa's third goal of the third period with 1:47 remaining in regulation to tie it 4-4. Elliott stopped 28 shots in his fourth straight start. Dany Heatley scored twice, earlier in the period, drawing the Senators to within one each time.
Chris Kelly also scored for Ottawa, which failed in its bid to win three games in a row for the first time this season.
"We showed a lot of resilience to come back and tie the game and get the point," Spezza said. "It would have been nice to win in the shootout, but we just have to move on."
Heatley scored his 18th goal 5:31 into the third to make it 3-2.
Kostopoulos put Montreal up 4-2 at 11:00 when he beat Elliott with a shot from the left circle for an unassisted goal.
Heatley brought Ottawa back within one just 33 seconds later with his 19th goal.
Robert Lang set up Kovalev's power-play goal 7:50 in, and earned his second assist of the period on Kostitsyn's 16th goal at 13:32, restoring Montreal's one-goal lead.
Ottawa, which entered with two wins in a row for the fifth time this season, has not won three straight since March 8-13, 2008.
"We're not worried about streaks right now," Heatley said.
Kovalev opened the scoring 7:50 in with his 12th goal, tallying during a 5-on-3 advantage. The Russian right wing was at the right side of the net to receive Lang's centering pass and put the puck into an open right side 27 seconds after Alfredsson was called for tripping while Spezza was in the penalty box.
Kelly tied it at 1 at 11:59 when he beat Halak into an open left side.
Kostitsyn extended his point streak to seven games as he took advantage of another fine setup pass from Lang to beat Elliott and put the Canadiens up 2-1 at 13:32.
D'Agostini beat Elliott with a quick shot from the slot 14:47 into the second after Canadiens center Gregory Stewart -- who made his season debut -- stripped the puck from Dean McAmmond behind the Senators' goal.