Bryan Little, with his 19th goal of the season, and Blake Wheeler, with his 23rd, scored for the Jets, who continued their winning ways under new head coach Paul Maurice. The Jets (29-26-6) are 10-3-1 since Maurice replaced the fired Claude Noel.
Oliver Ekman-Larsson and Martin Hanzal scored for Phoenix in the first game for both teams after a 19-day break for the Winter Olympics.
Jets winger Devin Setoguchi also beat Mike Smith in the shootout and Antoine Vermette was the only Coyote to solve Ondrej Pavelec in four rounds of penalty shots.
Winnipeg pulled within one point of the Coyotes (27-21-11) as both teams chase a wild-card playoff spot, although Phoenix has two games in hand.
Pavelec made 34 saves for the Jets, with two of his toughest stops coming in overtime. Smith stopped 27 shots for the visitors, including a pair of scoring chances in the final seconds of regulation.
Hanzal put away a rebound to tie the game at 2-2 with Jets centre Jim Slater in the penalty box 6:40 into the third period.
The Jets took advantage of a horrific giveaway by Coyotes defenceman Zbynek Michalek to take the lead late in the second period. With Ekman-Larsson in the penalty box, Little intercepted Michalek's pass in the low slot and beat Smith's glove hand to make it 2-1.
The teams wasted no time reminding fans of one of the differences between international hockey and the NHL game. Winnipeg's Zach Bogosian and Phoenix's Rob Klinkhammer dropped the gloves just 78 seconds into the game after Klinkhammer shoved Bogosian's defence partner Toby Enstrom into the boards.
The Coyotes opened the scoring with an Ekman-Larsson wrist shot from the high slot that went in off the post 12:17 into the first period.
It took the Jets less than three minutes to respond, as Wheeler's shot from the corner hit Coyotes defenceman Michael Stone in the leg and bounced into the net. Smith was distracted earlier in the sequence by Jets forward Dustin Byfuglien, and the Coyotes argued for an interference call.
Notes: Jocelyne Larocque of Ste. Anne, Man., a defenceman for Canada's Olympic gold medal-winning women's hockey team, performed the ceremonial puck drop. … The Jets welcomed back Slater, who missed 52 games with a sports hernia, and left-winger Evander Kane, who sat out the last six games before the Olympic break with a hand injury. … The Jets head to Nashville for a one-game road trip on Saturday, while the Coyotes play in Denver Friday night before heading home. … The teams have one remaining meeting, April 1 in Arizona.