Tomas Plekanec, Jordie Benn and Nathan Beaulieu also scored for Montreal (41-23-8), which moved four points ahead of the Senators in the race for the Atlantic Division crown.
Tom Pyatt scored for Ottawa.
Carey Price made 30 saves for the Canadiens while Craig Anderson stopped 33 shots for the Senators (39-24-8).
The Canadiens got off to a quick start in this one courtesy of a player who badly needed a goal. Plekanec, who has struggled to put up points this season, notched his first since Jan. 24 when he backhanded the rebound of Andrei Markov's point shot into an open side of the cage just 28 seconds into the contest.
Montreal kept up the pressure, immediately drawing a penalty following the goal and outshooting Ottawa 7-1 through the first 3 1/2 minutes of the game.
But the Senators finally found the offensive zone shortly thereafter. Pyatt, in the slot, picked up the loose puck off a Mike Hoffman shot and tossed it on net, hitting Benn's stick and deflecting past Price at 4:36.
Benn restored the Canadiens' lead late in the period when his blast from the point hit Viktor Stalberg and floated past a screened Anderson at 17:45.
Montreal continued its domination in the middle frame, outshooting the Senators 16-7 but couldn't add to its lead. And while rarely tested, Price kept his team ahead with an impressive tip of the pad save on Kyle Turris.
Byron connected for the sixth time in the last eight games to put the Canadiens up by a pair at 3:30 of the third period when he snapped home a shot from the right circle on a 2-on-1, reaching the 20-goal mark for the first time in his career.
Beaulieu made it 4-1 with a point shot that beat a screened Anderson five-hole.
Byron thought he had his second of the night a mere six seconds later but the goal was overturned after the Senators challenged for offside.
NOTES: Montreal C Torrey Mitchell played in his 600th NHL game. ... Canadiens C Tomas Plekanec appeared in his 912th NHL game, all with Montreal, moving into a tie with Guy Carbonneau for 11th place on the franchise's all-time games played list. ... The Senators iced the same line-up as Saturday night, leaving C Chris DiDomenico, D Jyrki Jokipakka and D Fredrik Claesson as the healthy scratches. ... Montreal made two changes to its lineup, with RW Michael McCarron drawing in for C Steve Ott while D Alexei Emelin returned after sitting as a healthy scratch, replacing D Brandon Davidson.