Celtic were dramatically denied a place in the fourth round after a penalty shoot-out defeat.
Henrik Larsson had pulled the side level on aggregate on the stroke of half-time, but the Spanish side won 5-4 in sudden death with Mista's spot-kick after Joos Valgaeren had his penalty saved - for the second time - after the referee had ordered a retake.
Lambert and Thompson scored easily for Celtic and Carew also kept his nerve. But Vicente's kick was saved before Larsson blasted his effort over the bar.
De los Santos brought the scores level, but Petrov fired wide. Sanchez put the Spaniards in front and Sutton equalised, before Ayala's penalty was saved to take it to sudden death. Hartson fired into the roof of the net and Pellegrino also scored, but Valgaeren missed at the second attempt.
The pre-match hype that suggested Valencia are in a state of some disarray seemed accurate early on as the Spaniards struggled to get any kind of grip on the game.
As early as the second minute they found themselves under intense pressure when Chris Sutton sent Larsson on his way, and although the Swede evaded the rash challenge of Santiago Canizares there was no way past Mauricio Pellegrino, who mopped up the danger and saved his goalkeeper's blushes.
It was the start of a frenzied attacking period from the Scottish champions and Valencia, who had Pablo Aimar booked for not retreating 10 yards at a free-kick, rode their luck.
Their unease was perfectly illustrated when Fabio Aurelio passed straight to Sutton. The striker in turn released Larsson, whose shot was blocked by Miroslav Djukic.
Having weathered the storm, however, the visitors began to unfurl a few sails, and their first attack worthy of the name required an outstanding save from Rab Douglas to deny Roberto Ayala.
There was no quarter asked or given, and Lubomir Moravcik was booked five minutes from the interval for a kick at Vicente that was well spotted by the Danish referee, Knud Eric Fisker.
As the break approached Celtic seemed to lose some of their discipline and Lubo Moravcik was booked for a late challenge on Vicente, whose first-leg goal still separated the two sides.
But just two minutes before the break the Slovakian had a glorious chance to put the home side in front on the night.
The referee allowed Moravcik to play on after Larsson was brought down and the Slovakian ghosted past Vicente, but blazed his effort just over the bar with Canizares to beat.
Then, in injury-time, Larsson scored with a majestic strike after being put clear by Joos Valgaeren for his 17th goal in Europe, surpassing Jimmy Johnstone's club record.
Valencia, with Jocelyn Angloma on for Curro Torres, began the second half purposefully and Celtic had three narrow escapes in quick succession.
First Vicente shot wide after sloppy defending, then Bobby Petta cleared a header from Gonzalo de los Santos off the line. Finally Aimar delayed too long on a golden opportunity and weight of numbers forced him to shoot past the post.
That was enough to encourage the Celtic manager, Martin O'Neill, to make a change, and after 55 minutes Momo Sylla replaced Moravcik just before a Ayala was booked for dissent.
Valencia again made another change with Ferrer Mista coming on for Fabio Aurelio in the 67th minute before Douglas had to get down well to save Vicente's powerful right-foot strike.
After 70 minutes Vicente became the third Spaniard to be booked after talking out of turn, and on 75 Angloma made it four yellow cards.
CELTIC: Douglas, Balde, Mjallby, Valgaeren, Petrov, Lambert, Lennon, Moravcik (Sylla 55), Petta (Thompson 85), Larsson, Sutton, Sylla (Hartson 111). Subs Not Used: Gould, McNamara, Tebily, Maloney. Booked: Moravcik. Goal: Larsson 45.
VALENCIA: Canizares, Ayala, Aurelio (Fagiani 67), Pellegrino, Carboni, Djukic, Torres (Angloma 46), Vicente, de los Santos, Aimar (Sanchez 83), Carew. Subs Not Used: Palop, Baraja, Salva, Marchena. Booked: Aimar, Ayala, Vicente, Angloma, Fagiani.
Referee: Knut-Erik Fisker (Denmark).