Senators defenseman Thomas Chabot scored a shorthanded goal from the point 52 seconds into the third to cut the Wild's lead to 4-2. Defenseman Christian Jaros made it 4-3 at 11:32 and forward Colin White tied it 4-4 at 12:55. It was Jaros' first NHL goal.
Craig Anderson made 28 saves for Ottawa (9-10-3), which allowed four goals or more for the sixth time in its past seven games.
"It just kind of feels like a waste," Senators forward Chris Tierney said. "You work that hard to get back and to give it up a couple minutes later is a sour feeling. You take a lot away; a lot of good hard work we did, but obviously we don't want to put it on ourselves to have to come back every game and score in the third. So yeah, it was a good effort, but in the end we gave them too much of a lead."
Wild goaltender Devan Dubnyk allowed four goals on 26 shots and was replaced by Alex Stalock after White tied the game. Stalock made one save in relief.
"I've never made that change in my life, and I don't want to make that change," Wild coach Bruce Boudreau said. "But I thought we needed a break and we needed something without calling a timeout, so you make the change and the change was long enough that I think it gave our guys a chance to calm down a little bit."
Greenway scored his third goal of the season when he one-timed a pass from Joel Eriksson Ek at 16:10 of the first period for a 1-0 lead. Dumba made it 2-0 with a slap shot on the power play at 19:39.
Staal deflected a pass from Granlund at 6:00 of the second period for a power-play goal to give the Wild a 3-0 lead.
Senators forward Mikkel Boedker found Tierney down low for the tap-in at 12:48 of the second period to make it 3-1. The assist was Boedker's 300th NHL point.
Dumba's 10th goal of the season made it 4-1 at 16:03. He extended his point streak to five games (five goals, two assists).
"I'm not trying to do anything too tricked up or special," Dumba said. "Guys are kind of getting it to me more often and finding my wheelhouse."
Wild forward Eric Fehr scored an empty-net goal at 19:22 of the third period for the final score.