The 19th-seeded Robredo, who came into the fourth-round match with an 0-10 record against the 17-time major champion, dismissed him 7-6 (7-3), 6-3, 6-4 on the Louis Armstrong court in New York after play was suspended for 4 1/2 hours by heavy rains.
Federer failed to reach the quarters of a Grand Slam tournament for the second straight time after his consecutive streak of 36 quarterfinal appearances was snapped at Wimbledon in July.
The Swiss icon's exit foiled fans' hopes of seeing him matched against fellow superstar Nadal in the quarterfinals -- the two have never faced each other in the U.S. Open.
The Spanish star made it there without Federer, rallying after a dropped first set to dispatch 22nd-seeded German Philipp Kohlschreiber 6-7 (4-7), 6-4, 6-3, 6-1.
Nadal, who claimed his only U.S. Open title in 2010, improved to 19-0 on hard courts this year and comes into his match against Robredo with a 6-0 all-time record against his countryman.
Ferrer now goes on to face eighth-seeded Richard Gasquet of France, who got past No. 10 Canadian Milos Raonic, 6-7 (4-7), 7-6 (7-4), 2-6, 7-6 (11-9) in a 4-hour, 40-minute struggle.