The teams headed to the room all tied up again after the second, with Montreal holding an 11-9 edge in shots in the period.
In the final stanza, with Nick Suzuki still in the sin bin from his kneeing penalty late in the second, Drake Batherson scored to give Ottawa a 4-3 lead just 29 seconds after puck drop.
The Sens doubled their lead courtesy of Colin White, who scored at 11:30 of the third to make it a 5-3 game.
Austin Watson added an empty-netter with 4:04 remaining in regulation.
Allen made 26 saves in the loss.
The Canadiens' next game is on Thursday night, when they take on the New Jersey Devils at Prudential Center.