The Reds were made to work hard to earn a 1-0 victory against the Championship outfit at the City Ground.
Diogo Jota scored Liverpool's match-winner in the 78th minute.
Klopp's men spurned a golden chance in the first half when Roberto Firmino saw a shot saved by Forest keeper Ethan Horvath - despite having Jota for support.
What’s been said
The Liverpool boss admitted he was surprised Firmino did not tee-up Jota for what would have been an easy finish.
But he was pleased that the Reds advanced into the semi-finals where they'll meet Premier League title rivals Manchester City in a mouthwatering clash.
Klopp told ITV: “We could have and should have played better but they made it really difficult.
“In the first half, we should have scored the sitter. That's really unlike Bobby that he doesn't square the ball. Diogo is in a good position.
“That would have helped obviously but they didn't have a lot in the first half - long balls, counter-attacks, these kind of things.
“But the game stayed open. The atmosphere is a massive boost for them, we went through and I'm really happy.”
‘A proper striker’
Jota notched the winner when he showed excellent movement to get on the end of Kostas Tsimikas’ cross.
Klopp heaped praise on the desire the Portuguese showed.
He added: “It was not a massive chance. It was a difficult one to finish off, bring his leg in the position and come in front of the defender.
“It was an outstanding goal and he's a proper striker with the desire to put the body in. Absolutely great.”