The Reds looked set to drop two points at home to Merseyside rivals Everton on Sunday before Divock Origi scored a dramatic winner in the 96th minute.
Jordan Pickford failed to catch Virgil van Dijks miss-cued shot and Origi was on hand to head home to earn Liverpool all three points.
Klopp ran onto the Anfield pitch after the goal went in and celebrated the winner with goalkeeper Alisson.
But former England defender Danny Mills believes Klopps celebration was disrespectful to Everton.
He told BBC Radio 5 live: I think its absolutely shocking, what Jurgen Klopp does.
Ok, go down the touchline a little bit, come out of your technical area, celebrate with your staff on the bench – I dont have a problem with that at all.
Wow!!! Jurgen Klopp charges onto the pitch to celebrate with Alisson!
Liverpool score a bizzare LATE goal to win the Merseyside derby 1-0. Simply incredible! pic.twitter.com/7NM4P0pltV
— Soccer AM (@SoccerAM) December 2, 2018
But to run into the middle of the pitch, I think its disrespectful to Everton.
Yes, its a derby, its a big game, but you cannot run into the middle of the pitch to celebrate with your players. Last minute or not, it doesnt make any difference.
Klopp, meanwhile, was quick to apologise for his excessive celebrations.
I have to apologise because I didnt want to be disrespectful but I couldnt stop myself, the Liverpool manager said. It just happened!
The plan was not to run [onto the pitch to celebrate]!
When I chose to stop I was close to Alisson. He was surprised! It was the last situation of the game.
Both teams delivered an unbelievable fight. Both teams tried really to play football. No harsh things. The game was really good.