The Stars scored two empty-net goals in the final minute to win for the fourth time in five games. Edmonton has lost four in a row.
It was the first shutout of the year for Lehtonen and the 23rd of a decade-long career. He had to protect a 1-0 lead after Rich Peverley scored with 9:25 left in the second period.
Dallas did not put the game away until Tyler Seguin and Antoine Roussel scored in the closing seconds.
Edmonton is tied for the fewest victories in the NHL this season with four and in their first 20 games of the campaign the Oilers have given up 30 more goals than they have scored.
Devan Dubnyk stopped 24 shots in the loss.