Van Persie at double to send Arsenal through
The victory secured first place for Arsenal in Group F, and although Borussia trooped off at the end thinking they were eliminated, Olympiakos's surprise 1-0 win at Olympique Marseille handed the German champions the slimmest mathematical chance of surviving.
Arsenal, who have 11 points, will finish top no matter what happens in their final match at Olympiakos on December 6, but Marseille (seven points), Olympiakos (six) and Dortmund (four) can still qualify.
If Arsenal and Olympiakos draw, and Dortmund beat Marseille, the trio behind Arsenal would all finish on seven points and the scores will determine who advances.
There is no doubt about the prowess of Van Persie who took his scoring record this year to 38 goals in 41 matches for Arsenal, with 17 goals in 18 appearances this season.
"He is in a period where he does not have to look for goals they just come to him, he is always in the right place at the right moment," Arsenal manager Arsene Wenger told reporters.
"And it looks easy but it isn't - he is there because he understands the game so well."
Dortmund manager Jurgen Klopp added his praise.
"Van Persie's performance tonight was just unbelievable, he is just such a player," he said.
Van Persie also scored against Dortmund when the teams drew 1-1 in Germany in their first match in September.
"I am not sure of he is the best striker in Europe at the moment, but he is among the best. I have hardly ever seen a player who plays so deep in midfield and is still so dangerous in the box," Klopp said.
The Dutchman maintained his incredible run when he headed home after a superb run and cross from Alex Song after 49 minutes.
Song darted between two defenders and provided the perfect cross which had Wenger fulsome in his praise.
"That was exceptional skill from a defensive midfielder," he said.
Van Persie, who gave Dortmund goalkeeper Roman Weidenfeller an easy save from a header in the first half, powered the ball downwards with such force that even though Weidenfeller got a hand to the ball, he was unable to stop it crossing the line.
With very few clear-cut chances for either team, that goal looked like settling the game until Van Persie made it 2-0 four minutes from time.
Left alone and unmarked at the far post, he swept in his second from close range after Thomas Vermaelen flicked on a corner from Mikel Arteta.
Shinji Kagawa got a consolation for Dortmund with the last kick of the game they started brightly, but gradually allowed Arsenal to control.
Apart from a couple of early attempts and one half-chance in the second half, Dortmund never looked like ending Arsenal's long unbeaten home run in the group stage of the competition.
However, their game plan was disrupted midway through the first half when they lost midfielders Sven Bender with a broken jaw and Mario Gotze with a knee injury within five minutes.