The five-times European champions had overcome the loss of captain Steven Gerrard, who lasted 24 minutes in his comeback game before limping off injured, to take a 41st minute lead through Yossi Benayoun.
But Lyon substitute Maxime Gonalons headed a 72nd minute equaliser before Delgado's late winner stunned Rafa Benitez's side.
"It was a deserved victory, we deserved what we got," Lyon coach Claude Puel told reporters.
"There's still lots of work to do though. Obviously Liverpool are going to come to our place to win because they have got to win.
"But we're going to savour the moment.
"Liverpool were strong to start with and put us under a lot of pressure, but we managed to remain calm and strong at the back."
Liverpool are in danger of dropping into the Europa League after falling three-points behind second-placed Fiorentina in Group E.
Lyon have a maximum nine points with Fiorentina on six, Liverpool three and Debrecen yet to register a point after three games.
TORRES MISSING
Liverpool were without influential striker Fernando Torres for the second successive game due to injury and Benitez rued the absence of the Spaniard.
"We had too many problems before the game with injuries," he said.
"But we still managed to take the lead and we still managed to create chances."
Liverpool looked in control after Benayoun's fourth goal of the season four minutes before halftime.
Substitute Fabio Aurelio, Gerrard's replacement, delivered a cross into the Lyon danger area which missed everyone before eventually falling for Benayoun to slot home.
From struggling to carve out a chance, Liverpool almost doubled their lead moments later when 19-year-old Martin Kelly, making his first senior start in the absence of injured England fullback Glen Johnson, delivered a pin-point cross for Aurelio who was denied by a magnificent save by Hugo Lloris.
Lyon turned the game on its head with two goals in the final 12 minutes thanks to two substitutes.
Gonalons headed the equaliser after Liverpool keeper Pepe Reina had pulled off two superb saves before Delgado hit a dramatic winner from Sidney Govou's cross.
Liverpool have now lost four successive games, two in the Champions League and two in the Premier League, and the final whistle was greeted by a chorus of boos.
