Henrikh Mkhitaryan had put the well-drilled Germans ahead in the 16th minute of the Group F clash at the Emirates Stadium after Gunners midfielder Aaron Ramsey had given away possession on the edge of his own penalty area.
However, Arsenal were level just before half-time through Olivier Giroud before substitute Santi Cazorla curled a 20-yard effort against the woodwork.
All of Arsenal's efforts, though, were undone as Dortmund - last season's beaten finalists - scored a winner with just eight minutes left with Robert Lewandowski rounding off a counter-attack.
It means Dortmund, Arsenal and Napoli are all level on six points in the group, and while Klopp admitted the Gunners created more, he pointed out his team had achieved their target of victory.
In quotes on Dortmund's official website, Klopp said: "I think it was not an undeserved win, but of course you need a bit of luck to win here in London.
"There are a few things we could improve on. But there have been games when we had 30 shots on goal and none went in.
"This time we had something like three shots, and scored two goals.
"The biggest step in development was that we struck eight minutes from time and are back in business again.
"We are not thinking about whether we are a top side, but how we can beat a top side.
"My team battled hard in every situation. Arsenal had the better chances, but we didn't come here to dominate Arsenal but to get a result."
Klopp, who watched from the stands because of a touchline ban, added: "We knew we had to be very disciplined.
"We were good at pressing and caused them problems.
"When Arsenal had space they moved the ball well and we were made to work hard. We were lucky when (Mats) Hummels cleared a ball off the line, and overall we had some ups and downs.
"But I always thought that we were capable of scoring the decisive goal if they give us the space.
"You need some luck, but the goal we conceded was also unlucky. It was not a dangerous situation.
"It was a game between two teams who kept up a high work rate and played with a lot of discipline at the highest level."