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 Sky Sports: "I was very impressed with our focus from the first to the last minute.
"It was a very tight game, very difficult. We didn't give them room to create chances, but then we had a difficult period at the start of second half.
"Nevertheless, we created chances and we could have scored a second. It was a very mature performance."