Goals from Roberto Firmino, who also missed a penalty, and Mohamed Salah appeared to have put them in control after a Dejan Lovren mistake had allowed Wissam Ben Yedder to score after just five minutes.
But with 18 minutes to go the defence switched off and Joaquin Correa nipped in to equalise with only their second shot on target.
To compound Liverpool's frustration Joe Gomez was sent off in added time for a second bookable offence.
It had all looked so good for an hour and even Philippe Coutinho, whose head was turned by the thought of joining Barcelona in the summer, must have been impressed watching from the bench.
It provided the Brazil international, who a month ago handed in a transfer request, with a reminder that while his current employers may still have some ground to catch up on the Catalan club when they are at their best they are one of the most exciting sides in Europe.
Coutinho was eventually sent on for the final 15 minutes but he could not find a moment of magic he has so often provided.
If Liverpool's success is based around the pace provided on either flank by Sadio Mane and summer signing Salah their downfall is almost always mistakes in defence
Mane, who starts a three-match domestic ban against Burnley on Saturday, was in scintillating form but his fellow winger was not far behind.
The Senegal international was a menace to Sevilla down Liverpool's left and in addition to the penalty he won, which Firmino crashed against the post, he created a number of other chances.
It was much-needed too as another defensive error put the hosts behind after just five minutes when Lovren, recalled to the side after sitting out the 5-0 hammering at City, fell over allowing Sergio Escudero's cross to roll under his foot and give Ben Yedder a simple finish from five yards.
However, Liverpool had not waited 1,009 days to let such a setback derail their return to the Champions League and they upped the tempo as Mane forced one low save from Sergio Rico from Salah's quick counter-attack and Firmino headed onto the roof of the net before the equaliser came.
Alberto Moreno exchanged passes with Jordan Henderson, whose intricately-threaded through-ball saw the left-back cross for Firmino to sidefoot home.
Salah has taken a while to adjust to Klopp's demanding work ethic but he was rewarded for his endeavour in chasing back to dispossess Steven N'Zonzi when his resulting shot deflected off Simon Kjaer and over the diving Rico.
Mane's claims for a handball against Nicolas Pareja went unheeded but he was immediately brought down by the defender only for Firmino to send the goalkeeper the wrong way but hit the upright.
There was no let up from Klopp's side after the break and Sevilla coach Eduardo Berizzo's attempts to slow things down by tossing the ball away at a Liverpool throw-in for a second time saw him sent to the stands.
Firmino, Salah and Georginio Wijnaldum all had chances to extend the lead but they were made to pay when they switched off from a throw-in and Correa burst through the middle to equalise.
Coutinho, plus Daniel Sturride's long-awaited Champions League Liverpool debut, failed to turn the game back in their favour.
In added time Sevilla substitute Luis Muriel missed a chance to snatch victory before Gomez was dismissed.
Source : PA