The German champions swept aside Shakhtar Donetsk 3-0 at the Westfalenstadion on Tuesday night to go through 5-2 on aggregate. Goals from Felipe Santana, Mario Gotze and Jakub Blaszczykowski sealed a convincing win for the hosts.
Klopp said: "This is an extraordinary moment, our whole Champions League campaign has run like clockwork. For 80 minutes it was an unbelievable game from us. Just after the break it was a bit concerning for a moment, but we were strong from the first minute to the last."
It is the first time since the 1997/98 season that Dortmund have been in the last eight of Europe's top club competition, when they eventually lost to Real Madrid in the semi-finals.
Chief executive Hans-Joachim Watzke added: "It's been 15 years since we were in the quarter-finals. It is about honour for us. It's good for us financially, but the feeling to reach the quarter-finals is irreplaceable."
Winger Kevin Grosskreutz claimed Dortmund's run would not end at the quarter-finals.
"We are overjoyed, everyone can be proud of themselves," he said on the club website. "In the quarter-finals we can again achieve a great sensation."
Shakhtar coach Mircea Lucescu admitted his side were outclassed. "Borussia deservedly progressed further," he said on the Ukrainian team's website. "They were the team that were hard to beat. I want to note once again that it was a well-deserved victory.
"We had hope up to the 30th minute, as long as the opponents didn't create anything, with us waiting for our counter-attacking chance.
"But, unfortunately, they scored a goal from a corner kick, and then all our plans changed. There were seven fatal minutes - from the 30th to 37th when the opponents scored again after our errors."