For the fourth time this season the Wild have won back-to-back contests and they will go after their first three-game winning streak on Saturday at Calgary.
Ryan Smyth put the Oilers in front midway through the second period, but Cullen tied it with 4:58 to go in the stanza.
Devin Setoguchi gave Minnesota the lead 2:20 into the third and Cullen finished the scoring with 6:01 remaining.
Niklas Backstrom collected the win with 27 saves. He had been out of action for a week with the flu. The Minnesota goals all came against Devan Dubnyk, who stopped 25 shots.