The second-seeded Nadal, who has lost just one set in winning his first five matches in this event, made it to the semifinals in his fifth consecutive U.S. Open appearance and will now take on No. 8 seed Richard Gasquet.
Gasquet survived a marathon match against fourth-seeded David Ferrer 6-3, 6-2, 4-6, 6-2, 6-3.
Nadal eliminated Robredo 6-0, 6-2, 6-2, taking another step in a bid to win his 10th tournament of the year. He has suffered only three losses all season -- in the finals of a tournament in Chile against Horacio Zeballos, in the title match against Novak Djokovic in Monaco and a shocking setback in the opening round at Wimbledon to Steve Darcis.
Gasquet reached a Grand Slam semifinal for the first time since Wimbledon in 2007. He has played Nadal 10 times and has lost every match.
Injuries kept Nadal out of last year's U.S. Open, but in the four years prior to that he compiled one championship, a runner-up finish and two semifinal appearances.
He is chasing his 13th Grand Slam title, eight of his previous triumphs having come at the French Open in a nine-year span.
Notable deaths of 2014 [PHOTOS]