Montreal has now reached the .500 mark after going 1-5-2 in its first eight games -- the Canadiens' worst start in 70 years.
"It was important to come out and have a good night," Price said. "When you sit around for a week, you have a lot of time to dwell on three wins, so I'm glad our guys knew to come in focused and reset.
Ottawa's lone goal came from Zack Smith early in the third period. Craig Anderson stopped 24 shots, dropping the club to 7-7-1. It's the second consecutive loss for the Senators, who had their six-game winning streak snapped with a 5-3 loss at Boston on Tuesday.
The play of Price and the work of his penalty-killers were the difference for Montreal. The Habs stifled the League's top power play, killing off all four Ottawa advantages.
"We got some key saves from the goalie at times," Montreal coach Jacques Martin said. "I thought our penalty killers did a good job. They worked hard and we know that (Ottawa) has great net presence and they move the puck around very well."
Ottawa coach Paul MacLean admitted that the lack of power-play production was a factor.
"We really could've used (a power-play goal) tonight," MacLean said. "Montreal did a good job of getting in the shooting lanes and not letting us get the puck to the net from the point, and we didn't did make the adjustments to get it to the net properly."
The Canadiens had the best chance late in the scoreless first period when a one-timer from Mathieu Darche deflected off defenseman Brian Lee's stick. The puck appeared ready to land in the crease, but Ottawa's Erik Condra caught the puck in midair and knocked it away.
Montreal finally opened the scoring at 13:55 of the second period when Cole took a feed from Mike Cammalleri and beat Anderson for his third of the season.
A giveaway by Erik Karlsson led to Montreal's second goal. Lars Eller picked up the giveaway, faked a backhand deke and passed to Kostitsyn, whose wrister past Anderson made it 2-0.
Karlsson, an All-Star defenseman last season, insisted the miscue was something he had to shake off.
"It was a misread from me and another player," Karlsson said. "He thinks I'm probably going to dump out or something and they capitalized on it. It's just the way it is sometimes. You just got to worry about the next shift, and right now, it doesn't really matter. We lost the game and if I can't take that, then I can't play in games here."
Ottawa got on the board 1:07 into the third period, when Zach Smith took a beautiful feed from Condra and beat Price with a backhander for a shorthanded goal.
The Senators had two third-period power plays -- including a late opportunity when Josh Gorges was called for holding with 1:31 left in regulation -- but was unable to capitalize despite pulling Anderson for a 6-on-4 skating advantage.
Price has allowed just five goals during the four-game winning streak, and Cammalleri said the goaltender has been a big factor in Montreal's revival.
"It was a balanced attack," Cammalleri said. "It was a four-line game. Everyone was trying to chip in. (The Sens) were trying – they were making some good plays to come back into the game. It was nice for us to hold them off. Carey was the biggest contributor to that. We needed a bit of a roll here after a real tough start. It's nice to get out of that hole."
Wrist shot -
1 - 0 MTL
Wrist shot -
2 - 0 MTL
Backhand shot -
2 - 1 MTL
Delay of game
Interference on goalkeeper