Yossi Benayoun had given Rafa Benitez's under-pressure side a first-half lead, but Maxime Gonalons and Delgado earned Group E leaders Lyon a victory to send tremors across Merseyside.
The Reds knew they were facing the prospect of four successive defeats for the first time since 1987 following setbacks at Fiorentina and Chelsea, and Saturday's loss-by-beachball at Sunderland, but they failed to produce a performance which merited a win.
Liverpool were rocked midway through the first half as captain Steven Gerrard was substituted as a precautionary measure following tightness in his groin, an injury which ruled him out at the Stadium of Light.
However, the Reds recovered and on 41 minutes the England ace's replacement, Fabio Aurelio, hit a cross-shot which fortuitously found Benayoun at the back post and the Israeli made no mistake.
The low-level confidence of Benitez's men was given a clear lift by the goal, but on 72 minutes the defensive frailties which have plagued their season were again evident as Gonalons nodded in following two good saves from Jose Reina.
Liverpool, who face Manchester United in the Premier League at the weekend, were then stunned on the stroke of full-time as Delgado tapped in following an electric counter-attack.
An air of nervous tension hung over Anfield, such were the levels of distress from fans unused to their side losing with such regularity.
Liverpool's form has been poor, a situation made worse by having their squad systematically deprived of its best players.
Fernando Torres was again out, while Glen Johnson was missing with a groin injury. At least Gerrard was back - but that lasted only 25 minutes.
Teenager Martin Kelly, Whiston-born and Academy produced, made his full debut at right-back, having had just a handful of minutes prior to this as a late substitute against PSV Eindhoven in Holland last December in this competition.
Facing them were a clever, experienced team, full of neat passing, solid possession and clever movement which threatened a fragile Liverpool.
It needed a close-range save from Reina to keep out a powerful header from Argentina striker Lisandro before patched-up Liverpool - who had David Ngog in Torres' role - continually wasted possession.
Former Liverpool and Lyon boss Gerard Houllier watched from the directors' box as the Reds struggled, while co-owner George Gillett was also in attendance, fresh from his trip to Saudi Arabia.
There were signs of Liverpool defiance, a Dirk Kuyt header disallowed for offside from Gerrard's deep cross.
That was virtually the only contribution from the skipper, who trudged off to be replaced by Aurelio. The groin injury sustained on England duty had given way again and facing United on Sunday looks a long shot.
It needed a performance of defiance, spirit and sheer desire from Liverpool to survive this, and there were occasional signs to warm The Kop but a clinical edge was missing.
Defender Cris was booked for a foul on Kuyt, the Dutchman's knee crashing into his head as he fell. The Brazilian did not last much longer, carried off in a daze.
Liverpool were beginning to show fight now, and somehow France No.2 goalkeeper Hugo Lloris kept out an Ngog effort after fine work by Kelly, Kuyt and Aurelio.
Aurelio had a powerful header saved by Lloris' left-hand from Kelly's run and cross, before Liverpool took a 41st minute lead.
Benayoun showed calmness to lift the ball over Lloris from eight yards, after Aurelio's cross evaded Kelly in the six-yard box.
Ngog was booked for a foul on Jeremy Toulalan, then Lyon skipper Sidney Govou was cautioned for flattening Emiliano Insua.
Benayoun, with a fine header, and Ngog - from eight-yards - could have stretched Liverpool's lead in the second period, while Lloris needed to be at full stretch to keep out Kuyt's header from Aurelio's cross.
Lyon sent on striker Bafetimbi Gomis to partner Lisandro in place of Ederson - and they began to stretch Liverpool.
Bosnian teenager Miralem Pjanic, followed relentlessly by Lucas, was starting to find space to produce the sort of tricks which have helped his country into the World Cup play-offs.
The pressure increased and Lyon equalised in the 72nd minute.
Liverpool failed to clear a corner, Reina made two outstanding saves from Toulalan and Govou before the ball reached substitute Gonalons, who dived to head home.
The hosts then lost young Kelly with an ankle injury, with Martin Skrtel taking over in a defence under increasing pressure.
Benitez sent on Andriy Voronin for the lively Benayoun, as the decision was greeted by boos from The Kop, as Liverpool searched for a second.
But the decider came in injury-time from substitute Cesar Delgado, arriving at the far post to sidefoot past Reina from Govou's cross.