Pierre-Marc Bouchard and Andrew Brunette both tallied in the third period for Minnesota, which has won six of its last nine games. Eric Nystrom collected two assists and Anton Khudobin made 31 saves.
"It really wasn't our best game, but at least we got the two points," said Minnesota forward Kyle Brodziak. "That's the most important thing, but we know that we must play better if we're going to stay competitive."
Nikolai Khabibulin gave up four goals on 26 shots for the Oilers, who returned home after a 1-4-0 road trip.
"It's a frustrating loss at home," said Edmonton forward Taylor Hall. "This is a place that we want to make teams scared to come into. With the exception of the first few minutes, we played a pretty good game. They put us on their heels at a time when we should be doing that to them."
Cullen backhanded a rebound past Khabibulin 1:39 into the contest, and Burns fired a wrister from the point that sailed through traffic.
Bouchard lit the lamp at 10:04 of the third period. Two minutes later, Brunette connected on a power play to make it 4-0.
Magnus Paajarvi spoiled Khudobin's bid for a second consecutive shutout with 3:46 left in regulation.
Edmonton forward Ales Hemsky left the game with a concussion...Minnesota has won all three meetings with Edmonton this season.