The 12-time Grand Slam title winner will face Richard Gasquet for a place in Monday's final after the French eighth seed clinched a rollercoaster 6-3, 6-1, 4-6, 2-6, 6-3 win over David Ferrer, the Spanish fourth seed.
World number two Nadal, the 2010 champion, took his hardcourt streak this year to 20-0 and his match record to an astonishing 58-3 with nine titles since his return in February after a seven-month injury layoff.
In a ruthless one hour, 40 minute demolition of Robredo, who had knocked out five-time champion Federer in the fourth round, the 27-year-old French Open champion took his perfect run of unbroken service games to 67 at the tournament.
He romped through the first set in just 22 minutes on Wednesday, allowing Robredo only five points, and was 2-0 up in the second before the 31-year-old got on the board.
Another break in the seventh game helped Nadal on his way to a two-set lead and breaks in the fifth and seventh games of the third helped put Robredo out of his misery.
It was the shortest US Open quarter-final since 1988, when Ivan Lendl beat Derrick Rostagno.
Nadal finished with 28 winners to Robredo's 10 and didn't give up a single break point as the world number two claimed an eighth win in eight meetings against his countryman, who has only won one set during that one-sided series.
"I am very happy to be in the semi-finals. I think I played my best tennis of this US Open. It's good to feel better with every match," said Nadal.
Gasquet, 27, whose only previous appearance in the last four at a major was at Wimbledon in 2007, is the first Frenchman to get to the semi-finals in New York since Cedric Pioline in 1999.
The victory was only his second in 10 meetings with Ferrer, a series which had also seen him claim just two sets out of 20.
Next up is Nadal, against whom he has lost all 10 meetings as a professional, although he did beat the great Spaniard in their junior days.
"I beat him when I was 13 in a junior tournament in Tarbes in France. Rafa is a good friend of mine so it'll be great to play him," said Gasquet.
Nadal said he remembered that 1999 defeat.
"I lost 6-4 in the the third set. It was a great match and was one of my first international tournaments. It was like the World Cup of junior tennis," he said.
"Richard played amazing that day — it was a great battle for me."
Gasquet had fought out the longest match of the tournament to see off Milos Raonic in the fourth round but showed no ill effects from that four-hour, 40-minute slugfest as he raced through the first two sets against Ferrer.
His majestic one-handed backhand was working overtime as Ferrer, a semi-finalist in 2007 and 2010, struggled.
But the 31-year-old Spaniard summoned up his trademark pugnacious instincts to claw his way back before Gasquet recovered with a crucial break in the sixth game of the decider when Ferrer double-faulted.
He took the match courtesy of a swinging forehand winner.
In a tight contest, Gasquet finished with 52 winners to Ferrer's 40 with the Frenchman collecting just five points more.
"I played a great match at the start despite the wind in the stadium. But David is a great fighter and I was a little tired after a big match with Raonic," said Gasquet.
"I got the break in the final set and it's an amazing feeling to be in the semi-finals. I have played a lot of five sets in my career but I still needed to play big to get the victory."
Ferrer admitted his sluggish start contributed to his eventual downfall.
"Richard played a very good game with his backhand, and was serving very good in important moments," he said.
"I tried to do my best. I fought until the last ball."
The two remaining quarter-finals will be played on Thursday when top seed Novak Djokovic, the 2011 champion who has not dropped a set so far, takes on Russian 21st seed Mikhail Youzhny, a two-time semi-finalist.
Defending champion Andy Murray, the third seed, meets Stanislas Wawrkinka, the ninth-seeded Swiss.