Demidov put Montreal up 3-2 when he buried the rebound off Alex Newhook's shot from the right face-off circle as he crashed the crease.
Juraj Slafkovsky and Alexandre Texier each scored for the Canadiens (36-18-10), who are 8-1-3 in their past 12 games, including a 3-1 win against the Toronto Maple Leafs on Tuesday.
Jacob Fowler, who was making his first start since Jan. 15 after being recalled from Laval of the American Hockey League earlier in the day, made 32 saves.
Drake Batherson scored twice, and Tim Stutzle had two assists to extend his point streak to 14 games (19 points; eight goals, 11 assists) for the Senators (32-23-9), who had their seven-game point streak end (5-0-2). Linus Ullmark made 20 saves.
Slafkovsky gave the Canadiens a 1-0 lead with a power-play goal at 1:16. Nick Suzuki sent a pass from the left boards to Demidov at the edge of the right circle. Demidov then waited out a diving Stutzle in the slot and fed a pass to Slafkovsky for a backdoor tap-in.
Batherson tied it 1-1 while on the power play at 16:42 when Brady Tkachuk's pass deflected in off his skate at the left side of the crease. The play was reviewed to see if the puck completely crossed the goal line, but the call on the ice stood as a good goal.
Batherson put Ottawa ahead 2-1 at 18:41. Claude Giroux knocked down Alexandre Carrier's clearing attempt along the left boards and passed to Batherson as he powered to the net. The puck came loose and Stutzle's shot at the right post hit Batherson's stick before going in.
Texier tied it 2-2 at 17:29 of the second period with a wraparound at the right post that went in off Ullmark's pad. He was in the lineup in place of Cole Caufield, who did not play after leaving a 3-1 win against the Toronto Maple Leafs on Tuesday because of an illness.
The Senators outshot the Canadiens 13-5 in the third period, but Fowler made multiple saves in the final moments of the game to preserve the one-goal lead.
