Roberto Firmino netted twice either side of a Sadio Mane goal to give the five-time European Cup winners a three-goal lead at half-time, before the hosts pulled two goals back within 15 minutes of the restart thanks to a brace from Wissam Ben Yedder.
Despite their wobble, Liverpool looked to have done enough to hang on to their lead, only for Guido Pizarro to snatch an equaliser in injury time.
While the result kept Liverpool top of their group it once again highlighted their defensive frailties, which have led to the Merseysiders conceding three or more goals in four occasions already this season.
"The description of the game is easy," Klopp told BT Sport.
"Fantastic first half for us, the second half we made a mistake – we didn't carry on playing football. It's normal to try to control the game but a team like us has to control the game with the ball.
"We stopped playing football in the second half and let them come back in the game."
Asked whether his team had been guilty of complacency, the German replied: "We were confident up to that first half and then we were too late in a situation, the whole formation was too late.
"They had the free-kick and the header and the penalty. The real problem is that we stopped playing football."
"We didn't play football any more. We became passive, they scored the first, then it was obvious the atmosphere changed immediately. After the second goal it was an open game again but we couldn't score off the counterattacks. We opened the door for them, didn't close it, so they could score in the last minute. That's the story of the game."
Despite their second-half capitulation, however, Liverpool remain in pole-position to qualify from Group E. A draw at Anfield against Spartak Moscow in the final game of the group stage will be enough to secure qualification to the following round, while a win will guarantee Klopp's men top spot in the group.