Coyle deked before beating Jets goalie Ondrej Pavelec with his second career shootout shot and first one that went in.
Minnesota is 15-5-4 and has won six of its past seven games and is 9-1-1 in November.
The loss extended Winnipeg' (10-11-4) winless streak to four games (0-2-2). It was the Jets' eighth shootout of the season and third loss in the contest.
Winnipeg was ahead 2-1 late in the third period and it looked like that lead might be secure when Wild defenceman Marco Scandella was called for hooking at 14:02.
But the short-handed situation was no challenge for the Wild.
Ryan Suter, Koivu and Parise somehow got on a 3-on-2 situation and Parise fired a low shot past Pavelec at 15:02.
Parise also had an assist and has 12 points in his last 10 games.
Pavelec stopped 32 shots in regulation and the Wild didn't get away a shot in the extra frame.
Backstrom faced 35 shots through regulation and four in overtime.
He was in goal for the Wild after Josh Harding suffered a lower-body injury in warm-up. One report said Harding slipped on a puck.
Wild backup netminder Darcy Kuemper was on his way to Iowa of the AHL, but was contacted in the nick of time at the Winnipeg airport and taken to the MTS Centre to sit on the bench in case Backstrom needed relief.
The Jets outshot Minnesota 14-4 in the scoreless period, with both teams each having one power play.
Frolik marked his sixth goal of the season late in the second period after the Wild turned the puck over just outside their blue-line.
Frolik got the puck and sent a pass to Halischuk, who fired high at Backstrom. The puck went into the air and then onto the ice near the crease, where Frolik was on the spot to flip it into the net at 13:07.
The Wild outshot the Jets 17-11 in the second period.
Minnesota tied it up 1-1 at 3:21 of the third when Scandella took a pass from Parise and fired a shot from the high slot. Niederreiter got the rebound and put it past Pavelec.
The Jets regained the lead just over two minutes later when Halischuk took advantage of his own rebound to score his second of the season.
Winnipeg embarks on a six-game road trip, beginning Monday in New Jersey. Minnesota finishes a four-game road trip in St. Louis on Monday.
Notes: Veteran Winnipeg defenceman Mark Stuart returned to the ice after missing the past 10 with a hip injury.