Price made 33 saves for the rested Canadiens, who had a six-day layoff before evening their record at 5-5-2.
"I'm not too surprised," Price said. "I was always confident in our group, I think we all were. We just weren't getting the breaks, we felt. We're all really confident in our ability so we just kept working at it."
Cole scored his third goal 13:55 into the second. Kostitsyn made it 2-0 with his fourth 3 minutes later.
The Canadiens' streak, which includes back-to-back wins over Stanley Cup champion Boston, began after assistant coach Perry Pearn was fired before a 5-1 win over Philadelphia on Oct. 26. Montreal's 1-5-2 start was its worst since 1941-42.
"We're happy with the two points," coach Jacques Martin said. "It's important. It puts us at .500 and because of our start, we've got some ground to make up."
Zack Smith spoiled Price's shutout bid with a short-handed goal early in third, but the Senators dropped their second in a row.
"We thought we had a good chance," Smith said. "I thought they were very desperate in their end tonight."
Ottawa's 5-3 road loss to the Bruins on Tuesday ended a six-game winning streak that followed a 1-5 start.
Craig Anderson stopped 24 shots.
Cole took a pass from Mike Cammalleri low in the left faceoff circle and snapped a shot past Anderson from a tight angle to give Montreal the lead.
The Canadiens took advantage of a turnover by Erik Karlsson to make it a two-goal advantage.
Pressured by Kostitsyn behind the Ottawa goal, the Senators defenseman threw a pass out front that Lars Eller cut off easily. The Montreal center drove in alone on Anderson before dishing to Kostitsyn for a shot into an open left side.
"Travis (Moen) and Andre did a great forecheck there and it created the space for me -- really good execution there," Eller said.
Smith got Ottawa within one 1:08 into the third, beating Price with a backhand after he was set up by Erik Condra.
"I said a few prayers in the penalty box there, promised a few things, so I better follow them up now," Gorges said.
The Senators, who entered with the league's top-ranked power play, failed to score and finished 0 for 4 with the man advantage.
"We had a lot of good looks," Ottawa center Jason Spezza said. "The only thing we're disappointed with in the power play tonight is not getting the results."
The sold-out crowd of 20,329 was the largest through eight games at Scotiabank Place. ... Price has posted three of his 12 career shutouts against Ottawa. ... Senators captain Daniel Alfredsson missed his third straight game because of a concussion. ... Ottawa G Alex Auld, who backed up Price last season, dressed as Anderson's backup after missing three games because of an undisclosed lower-body injury.