Olivier made it 1-0 at 15:47 of the first period on a shot from the right circle set up by Kent Johnson's pass from behind the net.
Philadelphia forward Olle Lycksell appeared to score a goal with 5:39 remaining in the second period, but it was overturned after a Blue Jackets coach's challenge determined the play was offside following video review.
Pyyhtia made it 2-0 at 7:05 of the second. Ersson went behind his net to play a puck, but it bounced over his stick and went to Columbus forward Sean Kuraly, who found Pyyhtia in front of an empty net.
After Frost scored his first goal, Cates tied it 2-2 at 8:48, banging in a loose puck in the slot.
Marchenko put the Blue Jackets back ahead 3-2 at 10:01 when he scored from in close.
After Frost's second goal, Christiansen scored his first of the season on a shot from the right point to give Columbus a 4-3 lead at 15:49.
"It's been a while, so it felt good," he said. "Nice play by [Dmitri Voronkov] and the goalie was pretty screened. You want to try to contribute to helping the team win. Felt good for a little bit there and try to focus on the next one."