Cale Makar scored twice, Andre Burakovsky had two assists, and Grubauer made 28 saves for Colorado (13-7-2).
"I didn't like our execution," Avalanche coach Jared Bednar said. "I just thought we had more than a handful of guys that just weren't good tonight. I think we worked and we competed. We tried to scratch and claw our way back into that, did a nice job in the second half of the second period earning enough chances to tie the game, but probably too little too late. You've got to play 60 minutes to win in this League, and not that we weren't trying to play hard, but I just thought decision making and execution was poor."
Zuccarello gave the Wild a 1-0 lead at 1:48 of the second period, completing a tic-tac-toe passing sequence with Mikko Koivu and Ryan Suter from the top of the crease on the power play.
Greenway scored 53 seconds later to make it 2-0, receiving a pass from Luke Kunin while cutting across the crease and shooting into an open net.
"I think we did everything we could tonight, but everyone did," Greenway said. "I think everyone played a good game."
Makar cut it to 2-1 with a power-play goal from the high slot at 14:19, and he tied it 2-2 at 18:49 on a shot short side from the right circle. Makar has 14 points (six goals, eight assists) in his past eight games.
"He's a difference-maker for us tonight again," Bednar said of Makar. "We had some horses tonight. I was overplaying a handful of guys because they were going. We need everybody going, especially with some of the guys we have out of the lineup. We need guys pulling their weight, and I don't think we had that tonight. We've had it, and tonight we just didn't. We'll review it and move onto the next one."