Sebastian Kehl scored a stunning extra-time winner to lift Dortmund past Hoffenheim 3-2 at Signal Iduna Park - three days after Klopp's men loss 1-0 to Bayern Munich in the Bundesliga.
Dortmund - last season's beaten finalists - took a 19th-minute lead via Neven Subotic but the home side were pegged back by Roberto Firmino and Kevin Volland.
Pierre-Emerick Aubameyang equalised 12 minutes into the second half before Kehl stepped up with the match-winner in the 107th minute.
Klopp was pleased with how Dortmund responded following Sunday's defeat at the hands of the German champions, while he also singled out Kehl.
"We were a tad better than the result suggests. It was an extremely intense match. We played well without having clear chances in the final third," Klopp said afterwards.
"In the second half we produced a superb reaction. We put an almost unbearable level of pressure on them and deservedly got the equaliser.
"And then Sebastian did that. Pulling something like that off today was quite something. I didn't know he was capable of that. It's brilliant that he gets to be the hero of the day.
"It was a dramatic game, which really swung back and forth. Following the defeat to Bayern and given the short-term absence of [Marco] Reus and [Mats] Hummels, the team reacted brilliantly today.
"Progressing to the semis is good for everyone. Now it's time to lick our wounds and look to the Gladbach match on Saturday."
The man himself, Kehl, was taken aback by the goal in Dortmund.
Dortmund dominated the first period of extra time, and were finally rewarded two minutes into the second period by Kehl's superb drive after Hoffenheim had failed to clear a cross into the box.
"It was an extremely emotional and aggressive match. I can actually hit a dropping ball pretty well. But I was a bit lucky that the ball flew like that and then hit the inside of the post," said Kehl.
"The DFB Cup now represents our only chance of silverware this season, and of course I still dream of making it to the final in Berlin once again. Let's see who we're drawn against in the next round. Hopefully it'll be another home game."
