Atletico's win over Sevilla last weekend had raised hopes of a revival after a woeful start to their La Liga campaign, but they were two goals behind and down to 10 men inside half an hour at a half-empty Nuevo Colombino stadium.
Bruno Fornaroli finished off a sweeping move to give the home side the lead on 14 minutes before Tomas Ujfalusi was shown a straight red card for a late challenge.
The visitors' misery deepened when teenage midfielder Cedric brought Aitor down in the penalty area and Jeronimo Barrales scored from the spot.
The result left Atletico, 11th in La Liga, on the verge of elimination before next week's second leg at the Calderon and provoked some sharp words from coach Quique Sanchez Flores.
"It's a huge disappointment," Sanchez Flores, who was brought in to replace the sacked Abel Resino at the end of October, said at a news conference.
"As long as nothing changes it's me who is responsible but at the same time I am going to look for solutions and try to find players who don't let me down," he added.
"This season I have had more disappointments than in my whole career up to this point."
Unai Emery's Valencia side, the 2008 winners, had the better of the early chances at the Mestalla, Joaquin and Juan Mata sending efforts flashing narrowly over the bar.
Mexican international Guardado stole in front of defender Hedwiges Maduro two minutes after halftime to steer the ball past Miguel Angel Moya and Alvarez doubled the visitors' lead with a fine run and finish.
Substitute David Silva pulled a goal back with a clever flick and Depor repelled late pressure to maintain the advantage for next week's second leg.
Defender Christian Fernandez was in the right place to head Racing into a first-half lead at Alcorcon's tiny stadium to the south west of Madrid.
Celta's bright start in driving rain belied their position of 18th out of 22 in the second-tier standings but they were rocked back by a Villarreal goal in the 12th minute.
