Zach Werenski and Dmitri Voronkov had a goal and an assist, and Adam Fantilli also scored in regulation as Columbus extended its point streak to five games (3-0-2). Jet Greaves finished with 29 saves.
Lane Hutson had a goal and an assist, and Josh Anderson and Oliver Kapanen also scored for Montreal, which has lost four straight (0-3-1) and six of seven (1-3-3). Jakub Dobes had 20 saves.
Kent Johnson also scored in the tiebreaker for Columbus, and Cole Caufield tallied for Montreal.
After Marchenko gave the Blue Jackets the lead in the third round of the shootout, Greaves stopped an attempt by Ivan Demidov to preserve the win.
Montreal outshot Columbus 6-0 in a scoreless overtime.
Trailing 3-1 after two periods, the Canadiens scored twice in the third to tie the score.
Anderson, cutting to the net, knocked the puck down and put it in from the left side at 8:28 to pull Montreal within one.
Hutson beat Greaves from above the left circle through traffic with 1:19 to go in regulation to tie it 3-3.
Fantilli gave the Blue Jackets a 2-1 lead at 9:47 of the second, scoring from in front.
Voronkov made it a two-goal lead with a power-play goal with 6:28 left in the middle period, deflecting Marchenko’s pass at the left doorstep.
Werenski got the Blue Jackets on the scoreboard first, scoring from the edge of the left circle 6:58 into the game.
Kapanen tied it for the Canadiens at 4:47 of the second, scoring from the slot off a pass from behind the goal line by Mike Matheson.
Canadiens: Host Washington on Thursday night.
Blue Jackets: At Winnipeg on Tuesday night.
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