Ciro Immobile opened his Champion’s League account by powering a shot past Laurent Kiscielny to beat Wojciech Szczesny with a low finish.
Dortmund doubled their lead when Pierre-Emerick Aubameyang finished another fine break away.
Arsenal need to improve significantly after a poor display at Signal iduna Park where they were outplayed for a better part of the game.
Arsenal were unbeaten in their last six visits to Germany and Dortmund were in no mood to let the run continue.
The German side had a total of 22 shots compared to Arsenal’s four and should have won by a larger margin after a sloppy and careless performance by Arsene Wenger’s side.
Arsenal were missing defenders Nacho Monreal, Mathieu Debuchy and Calum Chambers and fielded 19-year-old full-back Hector Belllerin, who was making his Champion’s league debut.
Sokartis Papastathopoulos and Henrikh Mkhitaryan had already let the Gunners breathe a sigh of relief with some wasteful finishes before immobile broke the deadlock in the 45th minute.
It came after Arsenal conceded possession on the edge of the Dortmund area, a long clearance finding former Torino forward Immobile who kept his composure before finding the net with a low angled finish after gliding past Koscielny.
Shortly before Immobile’s opener, Welbeck dragged a wonderful chance wide from inside the area after smart play from Aaron Ramsey.
It was to prove a costly miss and it went from bad to depressing for the Gunners.
Three minutes into the second half, Aubameyang skipped round Szczesny to bundle the ball over the line after the Gunners keeper and Per Mertesacker had got in each other’s way.
Aubameyang rattled the bar with a rising shot from the edge of the area after Arsenal’s defence was again exposed by another swift Dortmund attack.
‘They were very strong in midfield, stronger than us, I agree,’ said Wenger, who also defended Danny Welbeck, his new £16million signing who missed a clear chance to open the scoring.
‘He will improve,’ said Wenger. ‘I don’t think we have to make a problem of that. He had a lively first half, two or three good chances; one obvious one. We have to be a bit patient maybe.’
The Dortmund boss had this to say about his side’s performance:
‘I am very pleased,’ said Dortmund boss Jurgen Klopp. ‘Three players went out of the team because of muscle problems but we made some decisions and I have to say it was a great performance by the team. Really strong.
‘We were brave, we were fast and that is everything we needed for this game. We had other chances but two-nil is perfect for us against a strong opponent. I enjoyed it from the first second until the last. To see games like this, it is the best in the world. We play this quickly it’s very hard to defend against.’