Zack Kassian, Jack McBain, Lawson Crouse and Dylan Guenther also scored for the Coyotes and Clayton Keller had two assists. Connor Ingram stopped 31 shots in his first win this season for Arizona, which won for the second time in its opening six-game road trip.
''One of our big focuses was winning the first 10 minutes,'' Ingram said. ''Not having to play from behind was huge for my confidence.''
Johnny Gaudreau, Kent Johnson and Andrew Peeke scored for Columbus, which lost for the fifth time in eight games. Daniil Tarasov gave up four goals on 12 shots before being pulled in the second period for Elvis Merzlikins, who finished with five saves.
The Blue Jackets were 0 for 5 on the power play to fall to 0 for 20 on the season.
''We didn't have juice all night for whatever reason in the game,'' Blue Jackets coach Brad Larsen said. ''It really was from the goalie out. ... It just had a bad vibe to it kind of the whole game.''
Gostisbehere put the Coyotes on the board 24 seconds into a Columbus penalty at 9:24 of the first with a wrister from the right circle, his second power-play goal of the season.
Kassian made it 2-0 with 4:08 remaining with a double-tapped wraparound pushed under Tarasov's left skate.
''We capitalized on our chances, and then we played well defensively,'' Coyotes coach Andre Tourigny said. ''Even if they had a shot, we kept them to the outside pretty good in the first two periods and if you looked at the quality scoring chances, we were ahead there too. It was not the volume of shots for us. It was the quality.''
Columbus killed off a 5-on-3 situation in the second period to fire up the crowd, but Gostisbehere settled them down with his second goal of the night at 7:42 to make it 3-0.
Guenther's power-play goal extended the lead with 4:38 left in the middle period for his seventh point in six games. That was the last shot Tarasov faced as Merzlikins came in and gave up a goal to McBain at 2:10 of the third to make it 5-0.
Gaudreau put Columbus on the board at 6:45 of the third period, scoring off his own rebound in front of the net for his sixth point in seven games.
Johnson got his third goal in three games with 2:36 left, extending his point streak to four games.
Crouse had an empty-netter with 20 seconds left, and Columbus responded 11 seconds later when Peeke scored his second of the season.
The Blue Jackets welcomed back forward Patrik Laine, who missed six games with an elbow injury suffered in the team's Oct. 12 opener against Carolina, soon after he scored the club's first goal. Laine was expected to miss 4-6 weeks but came back after 13 days.
Gostisbehere's goal was his fifth power-play point of the season, which ties him for the league lead among defensemen and puts him third among all skaters. Arizona has eight power-play goals in six games.
The Blue Jackets remain the only team in the NHL without a power-play goal.
Justin Danforth suffered a torn labrum in the Blue Jackets' Oct. 22 loss to Pittsburgh. He is expected to undergo surgery Monday and be out for six months. Nick Blankenburg, who suffered an upper-body injury against the New York Rangers Sunday is day-to-day. The Blue Jackets recalled D Gavin Bayreuther, but he was a healthy scratch.
Travis Boyd's assist on Gostisbehere's goal was the 50th of his NHL career and extended his streak of helpers to three games. Keller's assist on the same goal extended his point streak to five games, the longest by a Coyotes player this season.
Coyotes: Host Winnipeg in their home opener on Friday night.
Blue Jackets: Host Boston on Friday night.
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