Boone Jenner extended the lead to 2-0 at 9:28, tipping in a shot from Roslovic on a power play.
Kopitar cut it to 2-1 at 4:42 of the second period over a sprawling Merzlikins.
Blake Lizotte tied it 2-2 at 8:25 before Laine restored the Blue Jackets lead on a 2-on-1 with a one-timer from the left circle to make it 3-2 at 10:03.
Jaret Anderson-Dolan tied it 3-3 at 15:02, scoring from the left circle after receiving a pass from Durzi.
"That's a good hockey team over there," Columbus coach Brad Larsen said. "It was a little bit of a messy game, as you saw, but that's a good team. They don't give up much. We were able to generate a little bit there. But they're in another zone really quick with the way they play."
NOTES:Gaudreau has 31 points (10 goals, 21 assists) in 27 games, tying Artemi Panarin (2018-19) for the second-fewest needed to reach 30 points in Blue Jackets history. Only Cam Atkinson did it in fewer games (26 games in 2018-19). Gaudreau has 13 points (three goals, 10 assists) in his past eight games. … Laine extended his point streak to four games (four goals, three assists). … Kopitar has 497 points (177 goals, 321 assists) in 621 road games, surpassing Dave Taylor for the second-most in Kings history. Only Marcel Dionne has more on the road (607 points; 262 goals, 345 assists). … Los Angeles defenseman Tobias Bjornfot had two shots on goal, one hit, one blocked shot and was plus-2 in 14:23 after being a healthy scratch for three games. … Columbus goalie Jet Greaves served as the backup after being an emergency recall Saturday.