Bayern gained revenge for their 5-2 humbling by Dortmund in last season's final with a battling display which was capped by Arjen Robben's superb, curling winner two minutes before half-time.
Dortmund improved after the break but rarely tested Manuel Neuer as Bayern eased through to the semi-finals.
"We served up a distinctly average first-half performance, that wasn't good enough," said Klopp.
"You need to be on your game for 90 minutes if you're to beat Bayern.
"The second half was okay, we played well and put Bayern under pressure.
"In games like this it comes down to the big moments, and Bayern made the most of theirs but we didn't. We didn't deserve to win today."
Bayern president Uli Hoeness felt the victory confirmed his club's current superiority over Dortmund - they also lead the two-time defending Bundesliga champions by 17 points in the title race.
"With this game, we have regained our dominance in Germany," he said. "We have a big advantage in the league and we fully deserved to beat them tonight.
"I think that resolves the issue of who is the strongest for this year."
Robben has been a peripheral figure in recent weeks but took his chance with both hands after replacing the suspended Franck Ribery in the starting XI.
"We want to win trophies. Today, we wanted to confirm we're in great shape," the Holland international said post-match.
"It was a game between two fantastic teams, not just in terms of Germany, but also in Europe.
"You have to be happy when you've scored the winner. We're very solid and well-organised.
"In the second half, we let them come at us a little too much, but we still had enough chances to have wrapped it up earlier."