The Senators faced a 3-0 deficit after Owen Nolan scored his 14th (power play) and 15th goal this season, both in Minnesota's three-goal first period.
Senators backup goaltender Alex Auld stopped 17-of-17 shots in relief of Brian Elliott, who was pulled after allowing three goals on 12 shots.
Ottawa scored twice in the third while a man down and came away with its first four-game winning streak since Jan. 4-12 of last season.
Minnesota got one goal from Pierre-Marc Bouchard and one assist from Nolan. Niklas Backstrom gave up three goals on 19 shots in the first two periods, before backup Josh Harding was beaten once on two shots in the third for the Wild, who have lost three of their last five outings.