The Reds had looked on course to confirm their qualification to the knockout stages from Group E but were denied victory by a second-half comeback with Guido Pizarro equalising in stoppage time.
Roberto Firmino gave Liverpool the perfect start in Spain, scoring after just two minutes, before a goal from Sadio Mane and another from the Brazilian had the Reds 3-0 up at half-time.
However, a second-half collapse saw Wissam Ben Yedder score twice to give the hosts hope before Pizarro equalised in the 93rd minute.
Klopp was critical of his side after the game, per BT Sport (h/t Football365.com):
"We stopped playing football in the second half.
"We have one real weapon – playing football – and we didn't do that in the second half before we conceded the second goal.
"It was absolutely OK that we were confident after that first half. It's obvious what we did – the real problem is we stopped playing football.
"Our main mistake was that for 15 minutes we didn't play football, we were passive, we were a little bit too deep. They fought back and well done to them.
"It feel like we lost but we didn't lose. There is a game to go and it's still in our hands but at the moment it feels really bad."
Albert Moreno will shoulder some of the blame having conceded the free-kick from which Ben Yedder headed his first goal and giving away the penalty for the Frenchman's second.
However, Jordan Henderson also put in a poor display, according to Tifo Football's Jack Lusby:
Jack Lusby @jacklusby_
Jordan Henderson has been much worse than Alberto Moreno tonight.2017-11-21 21:25:24
Football journalist Richard Buxton felt that complacency was to blame for the Reds' collapse:
Richard Buxton @RichardBuxton_
No matter who the manager or players, #LFC come unstuck time and time again due to sheer complacency2017-11-21 21:37:16
Liverpool's defensive frailties are well-known yet Klopp has been unable to rectify the problems that make his team so vulnerable.
While Klopp's team remain exciting going forward with Mane, Firmino, Coutinho and Mohamed Salah, the lapses at the back continue to undermine them.
Football writer Joel Rabinowitz said the game perfectly showed the best and the worst of Liverpool:
Joel Rabinowitz @joel_archie
If ever 90 minutes could neatly encapsulate the very best and worst of this Liverpool side, that was it. The capitulation felt inevitable at 3-1, which is the worst thing about it. All felt so inevitable.2017-11-21 22:08:04
Despite the result, Liverpool remain top of Group E, a point ahead of Sevilla with just one game left to play.
A draw against Spartak Moscow will be enough to secure qualification, while a win will send them through as group winners.
However, Klopp will be aware that Liverpool will need to seriously improve their defence if they are to make any kind of impression on the tournament this season.