The Canucks remained winless after four games and are the first team in NHL history to give up four straight multi-goal leads to start the season.
Johnny Gaudreau had a goal and an assist, Justin Danforth and Zach Werenski each scored, and Gavrikov added an assist before his goal at 3:21 of overtime on a pass from Yegor Chinakhov. Elvis Merzlikins stopped 32 shots for Columbus, which had lost its first three games.
''You got to find a way to spark your team,'' Blue Jackets coach Brad Larsen said. ''We needed to find a way to win one game and then go from there. Hopefully, that lets you take a breath, and it loosens up some guys.''
Elias Pettersson scored his 100th career goal and added an assist, and Bo Horvat scored twice for the Canucks. Spencer Martin stopped 28 shots.
Vancouver got on the board at 7:51 of the first with Horvat's short-handed one-timer off a pass from Pettersson with 13 seconds left in Curtis Lazar's roughing penalty. Pettersson followed at 11:11, burying a rebound from inside the right circle for his third goal in four games.
But as it has in every game this season, the Vancouver lead eventually evaporated.
Danforth put the Blue Jackets on the board at 15:42 of the second, beating Martin over the left shoulder, and Werenski pulled them even at 6:18 of the third, finishing a cross-crease pass from Boone Jenner.
''In the first period we didn't really have much,'' Werenski said. ''Both were desperate teams needing to win. I thought they came out a little bit better than us. The start of the second we started playing better and the third we dominated.''
Vancouver pulled ahead again 41 seconds later with Horvat's dribbler before Gaudreau skated in to beat Martin with a wraparound at 9:27 of the third.
''We competed,'' Vancouver coach Bruce Boudreau said. ''We came back. I thought in the last 10 minutes of the game we had 75% of the chances. We just didn't score.''
Boudreau remains one victory shy of becoming the 22nd coach in NHL history and the fifth active head coach to reach 600 regular-season wins.
NICK OF TIME
Nick Blankenburg made his season debut Tuesday after being a healthy scratch in the first three games of the season. The defenseman played seven NHL games after finishing his University of Michigan last spring, notching a goal and two assists. Adam Boqvist was a healthy scratch.
Columbus forward Kent Johnson celebrated his 20th birthday, playing 12:24 with one shot on goal.
The Canucks assigned defensseman Noah Juulsen to their AHL affiliate and activated defenseman Tyler Myers and forward Ilya Mikheyev from injured reserve.
Vancouver: Visits Minnesota on Thursday.
Columbus: Hosts Nashville on Thursday.
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