Three weeks after collapsing to a 3-3 draw with Anderlecht at the Emirates Stadium after holding a 3-0 lead, Arsenal went ahead in the second minute through Yaya Sanogo and made it 2-0 after 57 minutes through Alexis Sanchez.
This time, though, a similar self-inflicted implosion never looked likely.
Arsenal looked far more assured than they have done recently following two successive Premier League defeats and they inflicted a first Champions League defeat on Dortmund, who stay top of the group with a game to play.
Arsenal can still finish top if they win their last match at Galatasaray and Dortmund lose to Anderlecht, who, like Galatasaray are eliminated. Their main objective, though, was to seal a place in the last 16.
Arsenal boss Arsene Wenger, unsurprisingly, was a much happier man than after the Anderlecht collapse.
"Maybe we were defensively better tonight. We are a team that likes to go forward, sometimes we feel we dominate the game and we maybe forget to take precautions to defend. We did that better tonight."
The hosts needed only 73 seconds to take the lead with 21-year-old French striker Sanogo scoring his first goal for the club and the Gunners sealed victory when Sanchez curled in a wonderful strike from 20 metres.
It was Sanchez's third goal in five Champions League appearances in the competition proper for Arsenal, the same total he managed in 24 games for Barcelona and he has now scored 13 goals in his first 20 matches for the Londoners.
The result completed a treble of victories by English teams over German opposition this week with Chelsea beating Schalke 04 5-0 and Manchester City beating Bayern Munich 3-2 in their group games on Tuesday.
Sanogo, who limped off with a suspected hamstring injury just before the end, scored with a fine goal, converting after a neat one-two with Santi Cazorla, back-heeling the ball to the Spanish midfielder and then running on for the return.
Both teams went for goal throughout an entertaining game with Alex Oxlade-Chamberlain going particularly close after 55 minutes when his brilliant lob crashed back off the crossbar.
