With only a handful of chances, last season's runners-up Dortmund made the most of them, staying patient until the end to grab an 82nd minute winner through Robert Lewandowski to stun the hosts.
"The way we played those last minutes was just outstanding," said Klopp, who watched from the stands as he sat out a two-match suspension. "The team showed extreme maturity. The whole performance was very mature.
"We don't think about how we are a top team but how we can beat a top team. Arsenal had the better chances but we did not come to dominate, we came to get a result."
Dortmund by no means dominated the encounter in their usual all-out attacking fashion that saw them sensationally come back from seemingly lost situations in the competition last season.
Dortmund's youthful enthusiasm of two seasons ago, when they lost 2-1 to the London club, has now been replaced by a cool, methodical and disciplined head.
They have been equally efficient in the Bundesliga this season to stay one point behind leaders Bayern Munich.
Dortmund struck at just the right times to floor the Premier League leaders and bounce back with two straight wins after their opening Group F defeat by Napoli.
The win confirmed in the most emphatic way that Dortmund have lost none of their power and have also added depth in their squad with new signings Henrikh Mkhitaryan, who fired them into the lead, and Pierre-Emerick Aubameyang.
The Ruhr valley club was also able to brush off a string of injury absences, including defender Lukasz Piszczek, captain Sebastian Kehl and playmaker Ilkay Guendogan.
"There are things we can improve and there have been games where we had 30 shots on goal and still failed to score," said Klopp.
"In this game we had what felt like three shots and we scored two goals. The biggest development for my team was that we hit them eight minutes from time and now are back in the race."
- Sports & Recreation
- Robert Lewandowski