The Reds were handed their first Premier League defeat of the season on Thursday night.
Goals from Sergio Aguero and Leroy Sane cancelled out Roberto Firmino’s header as City ran out 2-1 winners at the Etihad.
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Liverpool’s lead at the top of the Premier League has now been cut to just four points.
A major talking point was Kompany’s tackle on Mohamed Salah, which saw the Belgium defender booked in the first half.
And Klopp was furious with Anthony Taylor’s decision, claiming Kompany had to be given his marching orders.
“I really like Vincent Kompany but how on Earth is that not a red card?” Klopp fumed.
“He is last man and he goes in. If he hits Mo [Salah] more he is out for the season. It is not easy for the ref and he may not see it how I see it.”
Liverpool’s lead at the top of the Premier League is now four points.
However, Klopp insists the Reds were unlucky to be handed their first league loss of the season, claiming Liverpool feel hard done by.
“It was a big pressure. Very intense game. We were unlucky in our finishing moments. Unluckier than City I would say,” he added.
“Sane scores and the situation with Sadio [Mane] when he hit the post. They had periods where they dominated the game and everybody felt the intensity.
“But we came back and had big chances. It is always like this. You have to score in those moments. When Aguero scores there is no angle. In similar situations we didn’t score.
“It was not our or City’s best game because we both made it difficult for the other team.
“I have already said to the boys this is OK. We lost it but it will happen. Tonight it is not nice but it is not the biggest problem.
“Our expectations are high. We can play better. On the other side it was an intense game. You have to take the game how it is.
“You can’t always dominate it. With a bit more luck we could have got 2-2, which would be perfect for us.”
Kompany was asked about the Salah tackle after the game, with the City star insisting it was a great challenge.
“I thought it was a great challenge, was it not?” he said.
“I got the ball, a little bit of the man but it wasn’t naughty. I didn’t try to injure him.
“It was that or let him go through on goal.
“I’m not worried about the result I’m buzzing with the tremendous performance.
“We went toe to toe with a very aggressive team and we matched them. It’s still in Liverpool’s hands.
“It came from our guts, a desire you can’t describe. We are a better team when we play with emotions.”
Liverpool are back in action on Monday against Wolves in the FA Cup.