The Armenian's goals were both excellently taken as he side-footed the first past Kevin Trapp and then rifled in a 25-metre effort from the edge of the penalty area early in the second half.
Dortmund's win put them top of the Bundesliga as the only team with 12 points from their first four games.
Defending champions Bayern Munich are two points behind in second with Hanover, Bayer Leverkusen and Mainz 05 on nine each. Eintracht are 15th with three points.
Mkhitaryan, who joined Borussia in July from Ukrainian champions Shakhtar Donetsk, where he orchestrated the side who reached the last 16 of the Champions League last season, opened his account 10 minutes into his third appearance.
Jakub Blaszczykowski threaded the ball through the Eintracht defence and Mkhitaryan, whose late father Hamlet was also an Armenian international, side-footed the ball first-time past Trapp.
Eintracht levelled before halftime when Takashi Inui floated the ball over the Dortmund defence, Aigner headed against the post and Michael Kadlec fired the rebound into an empty net.
The hosts started the second half brightly but were undone by another moment of inspiration from Mkhitaryan, who turned his marker, rode another tackle and unleashed a low shot from the edge of the area which flew past Trapp.
"Overall, we still have to play better football," said Borussia coach Juergen Klopp. "I have seen a couple of weaknesses in our buildup and, ideally, we need to defend better."
Klopp also complained about the Eintracht pitch as did Borussia goalkeeper Roman Wiedenfeller. "It hadn't been watered, it was rough and slow," he said.
Stuttgart gave former youth team coach Thomas Schneider a rousing start to his Bundesliga career after he replaced Bruno Labbadia who was fired on Monday.
Stuttgart were quickly off the mark as defender Ruediger opened the scoring and Ibisevic, who spent 4-1/2 seasons at Hoffenheim, headed the second from Maxim's corner in the first 20 minutes.
Kevin Volland pulled one back in the 26th minute but a blunder by Hoffenheim goalkeeper Koen Casteels allowed Stuttgart to almost immediately restore their two-goal advantage.
Casteels needlessly came out of his area to try to clear a long pass forward, missed the ball and Maxim scored into an empty net.
There was no letup for Hoffenheim after the break as Ibisevic struck again, then Maxim curled in his second from the edge of the area and the Bosnia forward completed his hat-trick with a header just after the hour.
Roberto Firmino pulled one back for Hoffenheim three minutes from the end as they suffered their first defeat of the season.