Andrew Ebbett, Antti Miettinen and Martin Havlat also scored for the Wild, winners of three straight overall.
Marc Pouliot and Ryan Potulny scored for the last-place Oilers, who finished a four-game road trip without a win.
The Wild's home winning streak over Edmonton is tied for the longest home winning streak against an opponent in the NHL, according to the Elias Sports Bureau.
Minnesota, which entered the night eight points behind Detroit for the eighth and final playoff spot in the Western Conference with 11 games remaining, has won three straight for the first time since Jan. 23-28.
The Wild have played well recently despite several key players missing games with injuries. Starting goalie Niklas Backstrom and defenseman Brent Burns are out. Also missing is role player Shane Hnidy and enforcer Derek Boogaard.
The Oilers have been plagued with injuries themselves this season. Edmonton players have missed 410 games with injuries, compared to 225 last season. The Oilers are 5-21-3 overall since Jan. 1, 1-15-1 on the road since mid-December and have a league-worst 49 points for the season.
Edmonton was looking for its second victory over the Wild in two weeks and was in decent shape when Potulny scored a 5-on-3 goal to tie the score at 2 early in the third period.
Kobasew put the Wild back in front 3 minutes later and Havlat then beat goalie Jeff Deslauriers a short while later to put the game out of reach.
Wild goalie Josh Harding was once again impressive in place of Backstrom. Harding stopped 34 shots and hasn't lost at home since Feb. 14, 2009. He's also stopped 113 of the last 118 shots he's faced.
NOTES: The Oilers assigned top prospect Jordan Eberle to AHL Springfield. Eberle, the 22nd pick in the 2008 draft, finished with 106 points this season for the Regina Pats of the WHL. ... The Wild signed college free agent center Casey Wellman to a two-year deal. Wellman just finished his sophomore season at Massachusetts where he scored 23 goals, sixth-most in Division I. Wellman participated in the pregame skate but didn't play. ... Minnesota also signed G Matt Hackett, its third-round selection in the 2009 draft, to a three-year deal. ... The Wild also called up D Justin Falk from AHL Houston as injury insurance.