The forward was traded to Dallas by the Carolina Hurricanes on Friday, and then signed an eight-year, $96 million contract.
Zach Hyman had two goals and an assist, Connor McDavid had a goal and an assist, and Leon Draisaitl had an assist to extend his point streak to 14 games for the Oilers (37-22-4), who took a four-goal lead into the third period and held on for their third win in four games. Stuart Skinner made 22 saves.
Wyatt Johnston, who signed a five-year, $42-million contract extension Saturday, extended his goal streak to five games for the Stars (41-19-2), whose four-game winning streak ended. Matt Duchene had two assists and Jake Oettinger made 22 saves.
Rantanen started the Dallas comeback with a power play at 5:06 of the third period to make it 5-2, scoring a one-timer on a Duchene pass to the left circle that he sent past Skinner's glove.
Just 11 seconds later, Jamie Benn made it 5-3 when he took a centering pass from behind the net by Evgenii Dadonov and sent a backhand past Skinner's glove.
Mathew Dumba made it 5-4 at 9:18. He sent a slap shot from the point that Skinner stopped but the rebound bounced in off of Oilers defenseman Ty Emberson.
Johnston put Dallas ahead 1-0 on the power play at 5:48 of the first period, picking up a rebound at the side of the net off a shot by Rantanen from the right-wing circle that was initially tipped by Jason Robertson.
Johnston has seven goals on streak, the second five-game run of his career.
Just 24 seconds later, Hyman tied it 1-1, tapping in a pass from McDavid, who had drawn Oettinger out the side of the crease before cutting back in and sliding the puck behind him along the goal-line.
Hyman made it 2-1 at 12:26, getting sprung on a breakaway off a long lead pass by Jake Walman, before sending a backhand trickling in under Oettinger's right arm.
Walman was acquired in a trade with the San Jose Sharks on Friday, and the defenseman became the fourth Oilers player this season to register a point in their debut game with the club.
Viktor Arvidson extended the lead to 3-1 at 17:42, sniping a one-timer far side past Oettinger from the right face-off dot off a cross-ice pass by Evan Bouchard.
Draisaitl had the secondary assist and has 21 points (11 goals, 10 assists) on his streak.
McDavid made it 4-1 at 5:20 of the second period, carrying the puck past Ilya Lyubushkin for a breakaway and sending a backhand high glove side past Oettinger.
Connor Brown made it 5-1 at 17:40, after his pass intended for Mattias Janmark on a two-on-one ended up being blocked by Stars defenseman Thomas Harley and going right back to him for the put in past Oettinger, who got caught sliding across his crease.
Dallas forward Roope Hintz left the game at 9:15 of the second period after getting hit in the face by a shot from Oilers forward Adam Henrique.