After a frenetic first half which produced little, Van Persie gave his side a 49th minute lead with a clinical header after fine work out wide from the inspired Alex Song, before the Arsenal captain doubled his tally from close range four minutes from time.
Shinji Kagawa pulled a consolation goal back for Dortmund in stoppage time, but it was an entirely efficient victory for the hosts as they sealed their place in the last 16 of the competition by topping Group F.
Arsene Wenger named an unchanged starting line-up as his side attempted to secure their passage through to the knockout stages with Laurent Koscielny continuing at right-back while the returning Abou Diaby was handed a spot on the substitutes' bench.
Starlet Mario Goetze, a reported £30 million target for the Gunners, started for the Germans, who needed a win to realistically keep alive their slim hopes of making the second round, and the midfielder was at the forefront of everything in what was a frantic opening.
Arsenal were forced to play on the break for much of the early stages, and Theo Walcott nearly punished the visitors as he surged through on goal to latch on to Aaron Ramsey's pinpoint through ball, but Roman Weidenfeller was swiftly off his line and made an impeccable challenge.
Dortmund coach Juergen Klopp had to make two enforced changes in the opening half hour as Sven Bender clashed with Thomas Vermaelen in innocuous fashion and had to be replaced by Moritz Leitner, while Goetze could not shake off an early knock with Ivan Perisic his replacement.
The visitors missed the dynamism of the two midfielders as Arsenal began to dictate much of the play, albeit without being able to muster more than one shot in the first half as the match as a whole lacked a cutting edge.
Aaron Ramsey was typically inventive in the midfield while Walcott posed a threat from the right, but Weidenfeller was frequently on hand to alleviate any danger in the Dortmund half as his side pressed incessantly.
Arsenal made a sluggish start to the second half as first Per Mertesacker was almost caught dallying on the ball by Perisic, before a slick one-two between Kagawa and Robert Lewandowski saw the Japanese midfielder fly through on goal, only for Wojciech Szczesny to make a fine stop with his legs at his near post.
But just seconds later, Arsenal went in front: Song distinguished himself as he jinked his way past two challenges out on the left and dinked a pinpoint delivery to the back post, where Van Persie was on hand to head past Weidenfeller with a clinical finish. It was his 16th Champions League goal for the Gunners.
Gervinho really ought to have doubled Arsenal's lead on the hour mark as he stole in behind the Dortmund back four to latch on to Ramsey's precise through ball, but he never looked convincing and dallied long enough to allow Mats Hummels to make a timely last-ditch challenge.
Lucas Barrios was introduced to further bolster Dortmund's attacking options midway through the second half as Klopp made his final substitution, but Arsenal defended stoutly to restrict the visitors to speculative efforts.
Van Persie capped yet another match-winning display with a close range finish after Vermaelen had expertly flicked on Arteta's inswinging corner, and his 38th goal in 41 games for the calendar year ensured his side all three points.
Kagawa pulled a goal back for the intrepid visitors in the final minute of play following an aberration from the hosts at the back, but it was nothing more than a mere consolation for the German side as Arsenal were left to celebrate becoming the first English team to seal their place in the last 16 this season.