Mohamed Salah was quick to restore parity for the Reds after Scott Arfield gave the visitors an unlikely lead at Anfield inside the first half-hour, but despite their domination of the contest from there on - with some 35 shots in all - Klopp's side were unable to find a winner.
"I'm not happy, I'm angry with the result but pleased with the performance," Klopp told reporters afterwards. "It's our fault. We were dominant, fantastic attitude. We played good, were fluent with seven changes [from the Sevilla game]. But only one goal.
"When you have shots on goal it's not about their defending. Their keeper made a few outstanding saves. I'm not happy, as when [Dominic] Solanke hit the bar I am sure it was a penalty.
"It's the best we have played against Burnley since I have been in. In the past we didn't feel good in the game. Today I saw fluid movements, passing between the lines, speed, crosses. We have to take the point and carry on."
Klopp's charges now go on to face Leicester City twice at the King Power, first on Tuesday night in the EFL Cup then again next Saturday in the league.