O'Reilly had missed two games with bruised shoulder but showed few effects of the layoff, netting the go-ahead score with 7:16 remaining in the third period.
Gabriel Landeskog also had a goal and Maxime Talbot provided an empty-netter with an assist as the Avs rebounded from Friday's home overtime loss to the New York Islanders.
Semyon Varlamov earned the victory with 25 saves.
Charlie Coyle set a Wild club record with two goals just 40 seconds apart in the second period, but it wasn't enough to extend Minnesota's four-game winning streak.
Niklas Backstrom stopped just 14-of-17 shots.