The Bundesliga outfit went into the Group D meeting at the Emirates Stadium already assured of their place in the knockout stages with four wins out of four but their perfect record was ended by goals from Yaya Sanogo and Alexis Sanchez.
That means Dortmund need to win at home to Anderlecht on matchday six to guarantee their place as group winners, although given their superior goal difference a draw will likely be enough.
Klopp had no complaints with the final result, conceding Arsene Wenger's side were good value for their victory.
"Over 90 minutes Arsenal were the better team," he said. "The positive aspects are no-one has hurt himself and we have not lost the head-to-head comparison [with Arsenal]."
Dortmund conceded inside two minutes to Sanogo, before Alexis' wonderful curling strike shortly before the hour mark sealed victory.
"That was the worst start that you can imagine," said Klopp. "But we were still in the game. There were a few situations where the last pass we played was not good.
"We were not courageous enough tonight. We did not close the gaps and we were not close enough to the players. There were a thousand reasons why we were not good enough tonight, but we saw that. There were no other explanations than that."
Meanwhile, midfielder Ilkay Gundogan echoed the sentiments of his coach.
"This wasn't necessary," he said. "We wanted to be very alert from the first seconds but we were overrun by Arsenal in the first 10 minutes.
"In the second half something was missing in our play. It was a deserved defeat."
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