Liverpool were on track to win an eighth straight game to start of 2018-19 thanks to Daniel Sturridge's acrobatic finish.
But Eden Hazard came off the bench to inspire a dramatic late turnaround as Chelsea struck twice in the last 11 minutes at Anfield.
Emerson Palmieri nudged home the equaliser before a phenomenal solo goal from Hazard booked Chelsea's passage into the last 16.
The Blues' equaliser survived a long VAR check for offside and Klopp also disagreed with the decision to give a foul for Naby Keita's tackle on Victor Moses in the build-up.
"There will be a lot of discussions about VAR, it's not perfect - offside, the free-kick for the first goal," Klopp told Sky Sports.
"The player clears the ball and it's a yellow card as well, the linesman is in a good position and it's not a free-kick.
"I cannot see it better than the referee but afterwards I think the first goal is offside, perhaps the VAR can see it not the referee.
"But three players are offside, two for sure offside and they block so they have impact.
"[Ross] Barkley - not a lot - but he's offside, they watched it and didn't think it was offside so we have to take that."
Despite being disappointed with the use of VAR, Klopp felt Liverpool should have killed the match off, with Sturridge hitting the crossbar with the scores level.
"We had the chances to finish it," Klopp added, with Liverpool set to visit Stamford Bridge for a Premier League rematch on Saturday.
"We hit the crossbar and had a few interesting balls, but they scored twice and we scored only once.
"We knew how Chelsea wanted to play but we gave them too much space and weren't compact enough. We still had the bigger chances in the first half.
"This is good preparation for Saturday but we have to defend better."