Elliott got his fifth career shutout as the Senators won for just the second time since the Olympic break (2-7-1). Ottawa also lost five in a row from Jan. 5-12 before putting together a franchise-record 11-game winning streak on its way to winning 13 of 15 prior to the break.
"It feels great, obviously the two points, but even more I think the way we played," said Senators captain Daniel Alfredsson, who assisted on both goals. "The way Elliott played was huge for us. I mean that gave us a lot of confidence, and I thought we played really well in front of him as well and really put a lot of pressure on them."
Peter Regin scored his first goal in 10 games midway through the first and Karlsson added a power-play goal in the third for Ottawa, which moved into sole possession of fifth place in the Eastern Conference, two ahead of Montreal and Philadelphia.
"We got traffic on the net today and we got pucks in there," said Karlsson, who went minus-four in Saturday's 5-4 loss in Dallas. "It's all a battle in front of the net and tonight I think we won them."
Jaroslav Halak stopped 26 shots in his eighth straight start for the Canadiens, who lost in regulation for the first time in eight games (6-1-1). Montreal ended its season-high six-game winning streak with a 3-2 shootout loss in Toronto on Saturday.
"[Elliott] played well, he made some big saves," Canadiens right wing Brian Gionta said. "We had some good opportunities where we could have got back into the game but it didn't go in."
Moen, who does not wear a visor, was cut by the blade of Cullen's skate when the Senators veteran was upended along the boards in the left corner of Ottawa's zone.
"I had no idea -- I just turned and saw him on the ice," Cullen said. "Oh, that's really scary. It leaves you with a sick feeling in your stomach, of course, and I certainly hope that he's OK"
Moen was trailing blood as he was rushed off the ice with trainer Graham Rynbend.
"It's scary when the skate comes up like that," Montreal defenseman Hal Gill said. "You just hope it's not the eye. You hope he's doing all right."
A Canadiens spokesman said that Moen was not expected to need to go to a hospital for any further medical attention.
Regin gave Ottawa a 1-0 lead 10:15 in when he redirected Karlsson's pass at the edge of the crease for his 10th goal.
With Montreal's Dominic Moore off for hooking, Karlsson gave the Senators some breathing room when he picked up his second point of the game with his second goal of the season 8:17 into the third. The rookie defenseman snapped a shot from the point through traffic and past Halak.
Canadian figure skater Joannie Rochette was on hand for the game. Rochette, who won an Olympic bronze medal in Vancouver after her mother died suddenly just prior to the competition, received a standing ovation from the Bell Centre crowd when she was introduced on the video board during the first period. ... Senators LW Milan Michalek missed his second straight game because of a knee injury. ... Elliott's last shutout was on Jan. 26 when he made 24 saves in a 3-0 win over Martin Brodeur and the New Jersey Devils. ... The Canadiens have been shut out seven times this season.