It was the first time that Dortmund have dropped points in the league this season, with Klopp forced to take off goalscorer Marcel Schmelzer and Marco Reus at the break due to injuries.
Per Nilsson then levelled matters six minutes after the interval and Klopp was frustrated by his side's defending early on in the half.
"After the break, we had to make two changes again and we had a weaker period in which we clearly did not defend well enough," he said.
"But we came back into the game well after conceding a goal and had some really great chances.
Klopp named Robert Lewandowski, Henrikh Mkhitaryan and Nuri Sahin on the bench for the visit to the Frankenstadion, after all three started in the 2-1 UEFA Champions League defeat against Napoli on Wednesday.
Despite the result, the Dortmund boss was pleased with the draw and believes that the result was fair.
"It was a fair result," he continued. "The Bundesliga is a tough business and my team fought well today.
"We did not want them to overwhelm us, so we tried to bring some freshness to the starting line-up.
"We are not unhappy with the point. It was an intense game with many duels."
Dortmund and Bayern are now level on points and goal difference at the top of the Bundesliga, with Klopp's side in first place on goals scored.
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