Forest had dumped out holders Leicester and Arsenal on their run to the last eight and could easily have caused another shock had the fine margins gone their way at the City Ground.
"It was a proper Cup match," said Klopp. "I know they don't get anything for it, but a massive compliment to them, they played a really good game."
Klopp made seven changes with Mohamed Salah, Sadio Mane and Andy Robertson among those rested and it showed in a ragged Liverpool performance.
Roberto Firmino should have put the Reds in front before the break when he tried to dink the ball over Ethan Horvath with just the goalkeeper to beat, but it was Forest who had the better chances after the break.
Philip Zinckernagel missed the best of them 14 minutes from time when he slotted wide Brennan Johnson's inviting cross.
Three minutes later, Jota showed the Forest forwards how it is done as he stretched to meet Kostas Tsimikas' cross for his 19th goal of the season.
Still, the home side could have taken the game to extra time as Ryan Yates headed straight into the arms of Alisson Becker and had a penalty appeal turned down as he rounded the Brazilian goalkeeper.
But Liverpool held on to book a meeting with City when one side's quest for history will come to an end.