Both beat goaltender Cristobal Huet while neither Chris Higgins nor Saku Koivu was able to score on Ottawa goalie Ray Emery. Ottawa coach Bryan Murray opted not to have struggling stars Daniel Alfredsson, Dany Heatley and Jason Spezza on his shootout list.
Sergei Samsonov scored his first goal as a Canadien with 3:23 left in regulation time to tie the game and force a scoreless overtime.
Ottawa (2-3-0) ended a three-game losing streak, all at home. Their only other win was 4-1 against the Toronto Maple Leafs in their home opener Oct. 4.
Higgins scored in the first period for Montreal (2-0-2).
Huet was sharp in goal for Montreal after a loose first period and Emery was strong for a second straight game for Ottawa as the Senators outshot the Canadiens 27-25.
Denis Hamel got the opening goal 7:07 into the game when he stripped Michael Ryder from behind in front of the Montreal net and slid a backhand that sneaked inside the post behind Huet.
Montreal tied it shorthanded at 11:24 when Koivu stole the puck from Spezza at centre ice, and sent Higgins in alone on the right side to beat Emery with a wrist shot.
It was Higgins' third goal of the season and his second shorthanded.
Kelly put Ottawa back in front when he banged in Patrick Eaves' pass across the crease at 13:49.
Samsonov tied it when checker Chris Neil lost his stick, allowing the Russian winger to circle in from the blue-line and beat Emery between the pads.
The Senators went 0-for-five on the power play and are now 1-for-30 in five games, while they have surrendered two shorthanded goals.
Ottawa has scored only nine goals this season. Heatley, who had 50 goals last season, and Peter Schaefer are among the Senators still looking for their first goal.
The Canadiens ended the game with only four defencemen as Craig Rivet and Mathieu Dandenault both left in the third period. There was no immediate word on their condition.
The highlight of the pre-game ceremony for the home opener was a standing ovation for head coach Guy Carbonneau from the sellout crowd of 21,273. Carbonneau, a former Canadiens captain, took over as head coach from GM Bob Gainey this season.
In player introductions, the loudest cheers were for captain Koivu, Huet, Higgins, star winger Alex Kovalev and rookie Guillaume Latendresse.
Notes - Alexei Kaigorodov made his NHL debut for Ottawa, taking tough guy Brian McGratton's spot on the fourth line. He got his first point, an assist on Kelly's goal ... Defenceman Joe Corvo (broken foot) remains sidelined ... Mark Streit and Garth Murray were scratched for Montreal.