Wyatt Johnston scored the winning tally in the shootout, beating Stuart Skinner’s glove with a wrist shot from the slot.
Johnston also had three assists, and Miro Heiskanen scored for the Stars (7-3-3), who extended their point streak to seven games (4-0-3). Casey DeSmith made 23 saves.
Connor McDavid had a goal and an assist, and Leon Draisaitl and Vasily Podkolzin each scored for the Oilers (6-5-4), who are 2-3-2 in their past seven. Stuart Skinner made 24 saves, and Jack Roslovic had two assists.
The Oilers took a 1-0 lead at 3:40 of the first period when Roslovic sent a one-handed pass across the crease to Podkolzin, who beat DeSmith with a quick shot from in tight.
Draisaitl made it 2-0 at 7:07 when he scored with a sharp-angle shot from the goal line to the right of the net on the power play.
Rantanen cut the lead to 2-1 at 9:10 of the second period, batting in a rebound after Johnston hit the post on the power play. It was also Rantanen’s 300th NHL goal.
McDavid pushed it to 3-1 at 7:51 in the third period with a wrist shot from the left face-off circle.
Rantanen cut the deficit to 3-2 with his second goal at 8:37 when he beat Skinner’s blocker with a shot from in front.
Heiskanen tied it 3-3 at 12:20. Rantanen took a shot that rebounded off the end boards directly to Heiskanen, who buried it into the wide-open net.