After a scoreless overtime, the game moved to a shootout.
Edmonton's Devan Dubnyk made a save on Jamie Benn to begin the tiebreaker and Eberle followed by jamming his shot past Lehtonen's skate and into the net.
The Stars' Rich Peverley missed on his second-round try and Perron went next, sending a shot over Lehtonen's glove.
Alex Chaisson lost the handle on Dallas' final attempt and the Oilers had their fifth win in seven games.
Garbutt netted a both goals for the Stars, who dropped a second straight in a shootout.
Lehtonen recorded 30 saves.