"He's a big body. If he learns to go to the net, he's going to get a lot of them," said Wild coach Bruce Boudreau.
Greenway's goal provided a bit of relief for the Wild, which outshot Arizona 20-8 in the opening 20 minutes of the game but escaped up just one goal on Kevin Fiala's opening tally at 14:39.
It was Fiala's fourth goal in six games as he, Mikko Koivu and Zach Parise again played a strong game. Parise would score Minnesota's second goal of the game 5:39 into the third period, giving the Wild a 2-1 lead.
"I feel like it's kind of been building that direction the last few games. I think we're building chemistry and learning," Parise said. "We haven't played a lot of hockey with Kevin so he doesn't know our tendencies and we don't know his. We are trying to learn and trying to become familiar with each other where we go on the ice and we can just hit it when we're under pressure.
"I think we're doing a lot better job using the back to the net and we're spending way more time in the offensive zone and now we're getting some scoring chances and that's the way it's supposed to be played."
The Coyotes, playing their third game in four nights -- all on the road -- again clawed back, tying the game once again on a goal by Logan Crouse 113 seconds after Parise's marker.
That set the stage for the Big Rig, who lit the lamp for the first time since last March 22, a span of 23 games without a goal.
Greenway has never been a prodigious scorer, but even that stretch of games was a little too lengthy for comfort.
But he said he never let that get him down.