David Desharnais scored in regulation and again in a shootout and the Canadiens beat the Senators 3-2 on Wednesday night to run their record for the season against their regional rival to 3-0-1.
"We just play well against them," said Montreal goalie Carey Price, who has been in the net for all four meetings. "We're a pretty skilled team and so are they."
Desharnais was the only scorer in the shootout, while Price stopped all three Ottawa attempts, including a diving poke check on Milan Michalek.
The Senators need to find an answer to the Canadiens soon. They will host Montreal on Friday night, and face them again March 23 in Montreal.
"We were happy to get a point on the road," Senators coach Paul MacLean said. "It was a little close to the vest on their part and on ours, but this team doesn't bring the best out in us, for some reason.
"We have another opportunity on Friday to find a solution. We don't seem to be able to skate like we do when we're successful. We have to find a solution to that."
Erik Cole also scored for Montreal (28-32-11), which is 3-0-1 in its last four games but remains well out of playoff contention.
Montreal wore its white "away" jerseys so that the Senators could get some exposure for their mostly black heritage gear, and it produced a game lacking in color or excitement for the fans. Ottawa outshot Montreal 37-26, but few big saves were needed at either end until the shootout.
A highlight came from Cole.
After scoring on his own rebound on a rush down the right side at 13:45 of the first period, the Canadiens' veteran inexplicably high-fived referee Greg Kimmerly in the midst of his celebration.
"I had no one to celebrate with," Cole said.
"That was hilarious," Price said. "I didn't laugh for long because they scored right after, but I think the ref was laughing, too."
When asked if the NHL might suspend Cole for making contact with an official, Price said: "You can't suspend someone for that. It was pure comedy."
Karlsson tied it at 11:01 when he threaded his 19th goal of the season through traffic and past Price from the top of the right circle. It was Karlsson's third goal in two games.
Montreal took a 1-0 lead on Cole's goal.
Only 26 seconds later, Greening was at the side of the net and finished a three-way play with Michalek and Jason Spezza to tie the game.
The crowd gave a warm ovation to defenseman Andrei Markov for his first shift on home ice in more than a year. Markov missed most of the last two seasons because of knee problems.
Montreal was without Alexei Emelin (upper body) and Scott Gomez (concussion). Tomas Kaberle didn't play because his girlfriend gave birth. Frederic St. Denis was called up from AHL Hamilton to fill in on defense. ... Canadiens center Tomas Plekanec missed warmups and got to the rink just before the start of the game because of family reasons, the team said. ... Bobby Butler, Zenon Konopka and Matt Carkner didn't dress for Ottawa. ... There were big cheers for a video tribute to former Canadiens captain Saku Koivu, now with Anaheim, who recently played his 1,000th NHL game.