Drake Batherson had a goal and an assist, and Matt Murray made 30 saves for the Senators (5-14-1), who are 3-2-0 in their past five games.
"I think Murray was our MVP tonight," Tkachuk said. "He's been our MVP for most of the year, or all the year, and now we've just got to help him out during those tough times as a group. He's been awesome for us."
Nick Suzuki and Corey Perry scored, and Jake Allen made 36 saves for the Canadiens (9-5-3), who have one win in their past five games (1-3-1).
"We were just sloppy in overtime," Montreal coach Claude Julien said. "We got tired at the end. That goal that they scored, we had two guys that had been out there for a long time and we didn't handle it well and we end up giving the opportunity to win the hockey game."
Suzuki gave the Canadiens a 1-0 lead at 1:17 of the first period with a wrist shot that went in off the right post on the rush.
"I think we're pretty much all up in our own heads right now," Suzuki said. "I think just overthinking it, playing not to lose, and that's never a good thing to do."