Bump tied it 1-1 at 10:24 when he redirected Emil Andrae's shot from the blue line off the goal post and under the crossbar. The goal was counted after video review showed the puck completely crossed the goal line.
The Flyers had chances to take the lead, including a four-minute power play that bridged the second and third periods, but they managed four shots on goal.
"The penalty kill won the game for us, really, especially that four-minute (advantage)," Blue Jackets coach Rick Bowness said.
Philadelphia, which entered the game last in the NHL on the power play at 15.5 percent went 0-for-4 with five shots on goal with the man-advantage.
"I think it hasn't been very good," Flyers defenseman Jamie Drysdale said. "I think we've had a lot of looks and haven't buried virtually any at this point. And I think when that happens you start to slow it down and look for perfect plays. Just wasn't clicking and it needed to."
After a scoreless third period, neither team had a shot on goal during overtime. Philadelphia had the best scoring chance of the extra period when Trevor Zegras sent Matvei Michkov in alone on Greaves, but a backchecking Adam Fantilli raced back to stick check Michkov and prevent him from getting a shot off with 26 seconds remaining.
"He can fly," Greaves said of Fantilli. "He was working back, and I saw him putting that pressure on so super happy he was able to make that play. Those moments are huge, and that's kind of what those games come down to."
Bowness said it's part of the commitment to the little details of the game that have driven the Blue Jackets, who are 15-2-4 in 21 games since he replaced Dean Evason as coach Jan. 12.
"They're committed to keep that puck out of our net, they're committed to the team play," he said. "(Conor) Garland had a tremendous backcheck in the third period that saved a goal. All those details, we preach details a lot, and we preach second efforts a lot, and that's the perfect example of it."
NOTES: Marchenko's 100th goal came in his 276th NHL game, making him the second-fastest Blue Jackets player to reach that number, behind Rick Nash (237 games). It also was his 25th goal of the season, after he scored 31 last season. Marchenko is the first Blue Jackets player with consecutive 25-goal seasons since Artemi Panarin from 2017-19. ... The Flyers went 1-for-1 on the penalty kill, ending a streak of six straight games with a power-play goal allowed.