On the winning goal, Panarin sent a backhand pass from behind the net to the right circle. The bouncing puck landed on the stick of Anderson, whose shot went off Gibson's left shoulder at the near post.
"I don't know if we played good enough to win but we played good enough to get it to overtime and get a point," Ducks forward Andrew Cogliano said. "It's embarrassing we let a goal up with a few minutes left. I was a part of that. It's absolutely unacceptable.
"Playing a good team like that you have to get it to overtime. It's embarrassing."
Columbus weathered a delay of game penalty on Cam Atkinson with 2:31 remaining in the third.
Anaheim pulled Gibson for a two-man advantage, and the Blue Jackets had two unsuccessful attempts at scoring into the open net before Nick Foligno scored his fifth goal with 29 seconds left.
The Ducks killed 6 of 7 penalties, including when they were down two men for 30 seconds in the third period. The Blue Jackets had five power-play shots for the game, and Anaheim had four shorthanded shots.
Columbus has two power-play goals in it past 33 chances.
Anaheim was 0-for-4 on the power play.
"We did some good things but we did some undisciplined acts that cost us points," Ducks coach Randy Carlyle said.
Adam Henrique, making his Ducks debut, helped Anaheim take a 2-1 lead in the second period. With the teams playing 4-on-4, his stretch pass sent Roy on a breakaway. Roy scored into the five-hole for his third goal at 2:32.
Panarin tied it 2-2 on the power play at 10:54 of the second.
The Ducks acquired Henrique, forward Joseph Blandisi and a third-round pick in the 2018 NHL Draft from the New Jersey Devils on Thursday for defenseman Sami Vatanen and a conditional third-round pick. Blandisi also debuted for the Ducks.
"I was just trying to adjust to the systems, different plays and different hockey language," Henrique said. "I felt pretty good."
Dubinsky put Columbus ahead 1-0 with his third goal at 7:02 of the first. Fowler's second of the season tied it 1-1 at 9:04 of the period.
