Van Persie scored for the 16th and 17th time in 18 appearances for Arsenal this season when he headed home after a superb run and cross from Alex Song after 49 minutes.
The Dutchman made it 2-0 when he swept in from close range in the closing stages. Shinji Kagawa got a consolation for Dortmund with the last kick of the game.
Dortmund had to win to stand any chance of advancing themselves, but apart from a couple of early attempts and one half-chance in the second half, they never looked like ending Arsenal’s long unbeaten home run in the group stage of the competition.
Dortmund’s game plan was disrupted midway through the first half when they lost midfielders Sven Bender and Mario Goetze within minutes of each other, and Arsenal gradually took control of the match and rarely looked in danger.
With three points their sole objective, it was not surprising that the German champions took the game to Arsenal almost from the kickoff with Arsenal’s Wojciech Szczesny the busier of the two keepers in the early stages.
Robert Lewandwoski and Shinji Kagawa had the first attempts while Arsenal were content to soak up Dortmund’s early forays, looking to hit the Germans on the break.
Theo Walcott forced goalkeeper Roman Weidenfeller to rush from his goal and clear after he had earlier made an easy catch from a Van Persie header.
However, as the opening half progressed the game became bogged down in midfield. But Dortmund’s game plan was hit midway through the half when, in quick succession, Bender and the highly promising teenager Goetze were both injured and replaced by Moritz Leitner and Ivan Perisic.
Arsenal took control at the start of the second half with Van Persie’s first goal and, although Weidenfeller got a hand to his powerful downward header, he could not stop the Dutchman’s effort. His second came minutes from the end when he swept the ball in after Thomas Vermaelen headed on.
Short link: