Mohamed Salah opened the scoring in the first leg of the quarter-final tie at Anfield on Wednesday, though the Egyptian was substituted early in the second half with a groin problem.
Alex Oxlade-Chamberlain and Sadio Mane increased Liverpool's lead in an astonishing opening half to the all-Premier League clash.
However, Klopp's squad have little time to celebrate the result as they have the small matter of a Merseyside derby against Everton on Saturday.
The German questioned the decision to book Liverpool an early kick-off time at Goodison Park, giving his injury-hit squad little time to recover from their European exploits.
"I can imagine a lot of people are happy about this, but we now play at 12:30pm on Saturday," a subdued Klopp told BT Sport .
"If you want to see games like this, then the Premier League have to think about it. Everton are perhaps happier about this game than we are, and that's not fair.
"Being in your national competition should not cause you problems. We now have the problem to somehow find players for Saturday when people are telling us it is one of the most important games of the year.
"I'm not really over the moon at the moment – I'm really angry about a few people who make decisions in this league.
"If we had all players available against Everton but I don't play the best XI, what happens then? We all know [what happens]."
Liverpool face a nervous wait to find out the severity of Salah's injury, particularly with the second leg against City taking place next Tuesday.
Klopp - who will be without Jordan Henderson at the Etihad Stadium due to suspension - revealed the forward had delivered a positive update in the dressing room after the final whistle.
"He said to me after the game that he's all fine," Klopp revealed. "However, I hope that is the case. We will have to wait."
Ragnar Klavan and Emre Can both sat out the home leg with injuries while defender Joel Matip will miss the rest of the season due to a thigh problem.