Liverpool suffered a fourth straight Premier League defeat as they were beaten 2-0 by their rivals on Saturday. It's also the Reds' fourth home league defeat on the bounce.
It was also the first time Everton won at Anfield in 22 years but the game did not pass without controversy.
Richarlison gave the Toffees an early lead before Gylfi Sigurdsson doubled their advantage from the spot with eight minutes to go.
Dominic Calvert-Lewin was adjudged to have been brought down by Trent Alexander-Arnold in the box.
Referee Chris Kavanagh was asked by VAR to look at the pitchside monitor, but the official took just a few seconds to decide he had made the right decision.
On the penalty decision, Klopp told talkSPORT: "It's not too cool because if I mention the penalty you know what kind of stories you and your colleagues will make of it.
"It is a situation when you look back the ref decided it. If the VAR calls you to the screen he has an obvious doubt. He didn't take long to stick to his decision and I can't comment on that.
"I'm not exactly sure what he saw, you'll have to ask him. If he didn't give the penalty we lose 1-0. It's not on me to point the finger at someone else.
"We have to keep playing, we have to keep working, fighting and we will definitely. We have to change the fortunes and that's the job now."
Kavanagh's a cursory glance at the pitch side monitor and stuck to his original decision raised a few eyebrows. The trend this season has been for the referees to change their decision once instructed to have a look at an incident again.
Former Liverpool midfielder Steve McMahon believes that the official did not handle things well.
"What they are saying to him is basically 'have a look and see if you can change your mind'. What they are saying is they don't think it's a penalty," McMahon told talkSPORT after the final whistle at Anfield.
"He goes over and glances at it and goes 'no, I've given a penalty. I believe it's a penalty'.
"Then if he believes it's a penalty, he then should take the next action, which I think is ludicrous, in sending him off.
"I don't advocate sending people off but if he's given a penalty he then must do the second thing by sending him off.
"He hasn't taken notice of this monitor. That's what it's there for. VAR is failing in lots of ways."
Jurgen Klopp believes the penalty awarded to Everton did not affect the outcome of the game at Anfield.