Aaron Ramsey scored his 11th goal for the season as the north Londoners moved to the top of Group F with their third win from four matches.
Dortmund controlled the game's first hour, with Henrikh Mkhitaryan and Jakub Blaszczykowski going close, while Marco Reus had an effort disallowed for offside.
Arsenal withstood Dortmund's best though – last season's beaten finalists had 13 shots on goal to the four of their opponents – before claiming the victory thanks to Ramsey's 62nd-minute header.
Wenger told reporters: "We were under pressure from first minute to last but we didn't make any mistakes and defended well.
"We were compact and worked together and although there was a difficult period in the first half we put in a very mature performance.
"I was concerned that there was a long way to go at 1-0 but we could have made it two.
"We had some good chances. We knew at half-time it was about keeping our focus and letting our technical ability win the game, which is what happened.
"Our defence was under a lot of pressure but they showed good awareness and tactical ability and that gives us a big lift. Not many teams come here and win."