Real were missing their two most expensive signings, Cristiano Ronaldo and Kaka, as well as several other key players, but were undone by a club with a 1.1 million euro annual budget compared to Real's 420 million.
It was the third defeat in five matches for Manuel Pellegrini's side and piled more pressure on the Chilean coach after Saturday's 0-0 draw at Sporting Gijon and last week's 3-2 home defeat to AC Milan in the Champions League.
"It's an embarrassment, I have no explanation," Pellegrini said at a news conference after the last-32 clash.
"When you lose like that there is not much to say. Sometimes in the Cup you think that your weaker opponent is easily beaten and until you lose you don't take them seriously."
Pressed about his future, he added: "I have never thought about, I am not thinking about, and I will not think about, resigning."
A fired-up Alcorcon, who play in the same league as Real's youth team Castilla, laid siege to the visitors' goal in the frantic early stages of the last-32 clash, forcing Jerzy Dudek into a couple of smart saves with shots from distance.
Former Real youth player Borja Gomez gave Alcorcon a shock lead in the 16th minute when he cut in from the right and curled a precise shot into the bottom left corner.
Spain defender Alvaro Arbeloa inexplicably clipped the ball into his own net six minutes later before Fernando Bejar headed against Dudek's left post.
Ernesto Gomez made it three five minutes before the break, sidefooting home Diego Cascon's cross from the right, and Borja pounced on a loose ball to make it four in the 52nd minute.
Pellegrini replaced Raul with Ruud van Nistelrooy with just under 20 minutes left as Real searched desperately for a goal.
Alcorcon keeper Juanma palmed the Dutchman's shot on to a post with four minutes remaining but Real were unable to find a goal ahead of the second leg at the Bernabeu in two weeks.
WOEFUL DISPLAY
Real director general Jorge Valdano apologised to the Madrid fans for the side's woeful display.
"It's difficult and I know how the Madrid supporters must be feeling," he said on the club's website.
"We have to take it as a lesson in humility that must serve as a starting point for a fresh beginning.
"We must stick together, and when I say together I am also thinking about the coach."
Real president Florentino Perez, who spent 250 million euros on players in the close season, was watching from the stands at Alcorcon's tiny stadium south west of the Spanish capital, which had its capacity boosted to 4,500 from 3,000 for the match.
In other Cup first legs on Tuesday, Quique Sanchez Flores marked his debut as Atletico Madrid coach with a 2-0 win at third-tier Marbella and Sevilla won 4-2 at Atletico Ciudad.
Another Segunda B side, Cultural Leonesa, play host to holders and la Liga leaders Barcelona on Wednesday, while last year's finalists Athletic Bilbao visit second
