The visitors took the lead in the 62nd minute with a header from Aaron Ramsey - the side's first attempt on goal - and went on to repel a late siege from Dortmund to secure a valuable three points.
Victory leaves Arsenal top of the group on nine points, while Dortmund - who were playing their 100th tie in the European Cup - slip out of the top two as a result of Napoli's win over Marseille.
Jurgen Klopp made one enforced change from the side that stormed to a 6-1 Bundesliga victory over Stuttgart on Friday, with Mats Hummels (thigh) making way for Neven Subotic.
Arsenal, meanwhile, were unchanged from Saturday's 2-0 victory over Liverpool that sent them five points clear at the top of the Premier League.
It was Subotic who had Dortmund’s first meaningful shot at goal, capitalising on a poor Aaron Ramsey clearance to hook narrowly wide of Wojciech Szczesny’s right-hand post.
Henrikh Mkhitaryan spurned the best chance of the first half eight minutes from the interval as he guided the ball wide from the edge of the penalty area having collected a through-ball from Jakub Blaszczykowski.
Szczesny was called into action early in the second half as Reus got on the end of a cross from the right and forced the goalkeeper into a diving save amid some questionable marking.
Reus had the ball in the back of the net a couple of minutes later as the hosts began to find holes in the Arsenal defence. Szczesny made a superb save from Blaszczykowski, before Reus pounced on the rebound only to be deemed offside by the assistant referee.
Klopp compared Arsenal's style of play to a symphony orchestra in the build-up to this match, stating a preference for his own brand of heavy-metal football. It was his side, though, who were left singing the blues as Ramsey put the London club ahead with their first chance of the match.
Mesut Ozil chipped a wonderful ball into the penalty area and Olivier Giroud got his head to it before Ramsey provided the decisive touch, nodding beyond Roman Weidenfeller.
The Welshman went close to doubling the visitors' lead four minutes later but was denied by an improvised Weidenfeller save, and Nuri Sahin was forced to clear off the line from the resulting corner.
As Arsenal suddenly stepped up a gear, Per Mertesacker was next to go close, cushioning a header narrowly over the Dortmund crossbar.
Robert Lewandowski headed over in the dying minutes, but Arsenal held on for maximum points.