Prior to his second-round loss at Wimbledon earlier this year, the 17-time Major winner had qualified for the last eight of 36 Grand Slams in a row, but his poor season continued as he was outplayed by Robredo at Flushing Meadows.
The Swiss maestro was error-prone from the beginning of the contest, dropping his serve in the opening game, but Robredo could not take immediate advantage as Federer levelled the scores at 2-2.
Each player would be broken once more during the opening set, which Robredo, ranked number 22 in the world, eventually took 7-3 on a tie-break.
The second set was also a tight affair, but the Spaniard once again came out on top on the crucial points, saving three break points before breaking Federer to love in the eighth game on his way to taking a two-set lead.
Robredo saved a further six break points in the third set and a seemingly deflated Federer then lost another service game without picking up a point to put Robredo on the edge of victory at 4-3.
The 31-year-old went on to close out the match, claiming his first career win over Federer in 11 matches courtesy of a 7-6(3) 6-3 6-4 scoreline. He could face Rafael Nadal in the quarter-finals.