Raanta made 25 saves and two huge ones that set the stage for the comeback. He stoned Jason Zucker all alone inside to keep the game 3-1 in the third period, just before Connauton took a Stepan pass and beat Minnesota goalie Devan Dubnyk with a sneaky wrist shot by the glove at 8:02 to bring Arizona within a goal.
Then in overtime, when the Wild had the puck for most of the first three minutes, Raanta had a huge save all alone against Zach Parise to keep the game going and set the stage for Keller.
The Coyotes, who allowed three goals in the first period at Winnipeg on Tuesday including one 30 seconds into play, had a much better start but Minnesota got the only goal.
Raanta made a nice pad save on Nino Niederreiter's bid to jam home the puck, but the Wild kept possession and Jared Spurgeon fed Eric Staal for his 24th goal at 8:04.
The Wild dominated the first 10 minutes of the second period and doubled the lead at 5:54 when Matt Cullen got behind Connauton, took a pass from Gustav Olofsson and beat Raanta with a backhander under the bar.
Just 4:04 later, Mike Reilly settled down a puck that popped up off the skate of Cullen and surprised Raanta with a quick shot low to the short side, suddenly the Coyotes were in a 3-0 hole.
Arizona had to change the momentum and Cousins provided a jolt with a big play.
Pulling the puck off the boards, Cousins threw a cross-ice pass to Goligoski and immediately sprinted to the slot. Goligoski gave the puck back to Cousins and he beat Dubnyk under the crossbar.
After two periods, the Coyotes trailed 3-1 but had more shots, shot attempts, scoring chances and face-off wins - and the Wild had allowed an NHL-worst 70 goals in the third period.
The stage was set. And the unlikely offensive hero of late got it started.
Connauton, who had one goal in his first 42 games, scored his third in four games since the All-Star Break and the Coyotes were within a goal at 3-2.
"We had some belief in the locker room and I think that's something we've been lacking," Connauton said. "I'm just trying to get pucks to the net and shoot every chance I get. Take more snap shots and get them away quicker."