The 17-time grand slam champion lost 7-6(3) 6-3 6-4 in a major boilover at Louis Armstrong Stadium, having converted only two of 16 break points and sprayed 43 unforced errors.
"That's amazing, hey?" Robredo, a gritty 31-year-old ranked 22 in the world, said court-side after advancing to his first quarter-final at Flushing Meadows.
"To beat the greatest player of all time, in a stadium like this, I'm delighted.
"Today the only difference was I did the break points and he did not. That is why I won the match in three sets."
Robredo missed a chance to serve out the first set, but pounced in the tiebreak, smacking a forehand winner to bring up three set points and closing it out with an ace.
The unfancied Spaniard, who has enjoyed his greatest grand slam success on the French Open's clay courts, cantered through the second set in 42 minutes, before breaking Federer's serve to love in the eighth game of the third set.
The 19th seed sealed match point with a big serve down the T to close out the match in an efficient two hours and 24 minutes.
Having slipped to seventh in the world in the lead-up to the tournament, Federer will finish the year without a major title after being dumped from the second round at Wimbledon and failing to surpass the quarter-finals at Roland Garros and the last four at the Australian Open.
