Phoenix was more sound defensively than in its first two games and created some good scoring chances because of it, sending 42 shots at Columbus goalie Steve Mason in his first start of the season.
Jason LaBarbera was sharp in his first action of the season, stopping 21 shots after starter Mike Smith left midway through the first period with a lower-body injury.
Sullivan had his eighth career hat trick, Ekman-Larsson his first three-point game and nine players had at least a point to help the Coyotes avoid their first 0-3 start since 1985-86, when the franchise was still in Winnipeg.
Fedor Tyutin scored for Columbus in its first regulation loss of the season.
A year ago, Phoenix surged to the Pacific Division title behind a tight-checking style and superb goaltending by Smith.
Neither was working through the first games this season, the defense giving up too many good scoring chances and turnovers, and Smith too many goals that wouldn't have gotten by him last season.
Phoenix allowed 10 goals the first two games, against Dallas and Chicago, and all those came against Smith, who began the day with a save percentage of .825.
The Blue Jackets put up plenty of shots against the Coyotes the last time these teams met, peppering Smith with 54 on April 3. He stopped them all, though, setting an NHL record for most saves in a shutout.
He didn't get a chance to match it, facing four shots before leaving the ice during a timeout.
LaBarbera sprawled to kick away a point-blank shot by Ryan Johansen right after coming in, then gave up a goal in the second period to Tyutin, who beat him with a big slap shot from near the blue line.
LaBarbera was solid after that, doing the splits to stop a couple of shots on a scrum in front of his crease in the second period and turning away seven more during a power play in the third.
The good defense helped Phoenix create some good offensive chances.
Sullivan scored his first goal with the Coyotes midway through the second period, punching in a rebound after Mason made a tough save on a shot by Antoine Vermette.
Ekman-Larsson made it 2-1 late in the period, beating Mason with a wrister from between the circles after taking a drop pass from Matthew Lombardi.
Columbus dug itself a deeper hole early in the third period, when Derek Dorsett and Derick Brassard were called for separate tripping penalties, giving Phoenix a two-man advantage for two minutes. Sullivan capitalized, one-timing a shot from between the circles on a feed from Mikkel Boedker past Mason.
Sullivan put the game out of reach near the midpoint of the period, squeezing a shot between Mason's pads on a redirect off a pass by Derek Morris. It was his first hat trick since Dec. 14, 2009 against Columbus while he was playing for Nashville.
NOTES: Coyotes C Martin Hanzal and D Rostislav Klesla were both out with lower-body injuries. ... Rocker Alice Cooper, a Valley resident and big Coyotes fan, attended the game. ... The Blue Jackets have failed to score on their 18 power plays against the Coyotes, including 0 for 3 Wednesday night. ... Sullivan has 48 points in 48 career games against Columbus.