Marchenko scored from the left face-off circle with a shot that went in off Rangers goalie Igor Shesterkin. It was his second goal of the game. He also had an assist.
Charlie Coyle had two assists, and Elvis Merzlikins made 26 saves for the Blue Jackets, who ended a two-game losing streak.
The Blue Jackets (30-21-8) are three points behind the Boston Bruins for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
The Rangers (23-29-8) scored four times in the third period but lost for the second time in three games since the Olympic break and for the fifth time in six games overall.
Gabe Perreault scored twice and had an assist. Will Borgen had a goal and an assist, and Vladislav Gavrikov also scored. Shesterkin made 23 saves.
The Rangers started their comeback with two goals in the first minute of the third period; Gavrikov made it 1-0 at 30 seconds, and 24 seconds later Perreault scored off a turnover to cut it to 4-2.
Columbus called a timeout after Perreault's goal, but the Rangers kept the pressure on and made it 4-3 on Borgen's goal at 12:50 off a netfront scramble with Shesterkin pulled for an extra skater because of a delayed penalty on Columbus.
Perreault then tied it 4-4 at 15:14, scoring off a pass from Borgen.
Adam Fantilli gave Columbus a 1-0 lead at 5:50 of the first period, scoring from below the right hash marks on a redirection of Marchenko's pass from the top of the left face-off circle.
Marchenko extended the lead to 2-0 with a power-play goal at 15:32.
The Rangers thought they made it 2-1 at 3:04 of the second period, when J.T. Miller appeared to score a power-play goal with a one-timer from the right circle.
The Blue Jackets challenged for goalie interference and video review determined Alexis Lafreniere made contact with Merzlikins' head prior to Miller scoring, so the goal came off the board.
Sean Monahan scored a short-handed goal 23 seconds later at 3:27 to give Columbus a 3-0 lead.
Mathieu Olivier made it 4-0 with a goal off the rush at 11:54.
Columbus defenseman Zach Werenski was a late scratch with an illness.
