The 10th-seeded Isner is also the first man from the U.S. to get to the final 16 at any Grand Slam tournament since Andy Roddick and Mardy Fish at the 2012 U.S. Open, although Fish withdrew before his fourth-round match.
Isner saved all 13 break points he faced against Robredo, a 2013 quarterfinalist at Roland Garros, where the Spaniard was the first man since 1927 to win three consecutive Grand Slam matches after dropping the first two sets of each.
There was only one service break Friday, and it came with Robredo serving at 5-all in the fourth set.
At 15-all, he double-faulted for the second time all match. Isner then hit a blistering forehand return winner to get to 15-40, and converted his sixth break point of the 3-hour, 18-minute match when Robredo put a backhand into the net on an 18-stroke exchange.
Isner - best known for winning the longest match in tennis history, 70-68 in the fifth set at Wimbledon in 2010 - then served out the victory at love. He finished with 17 aces and ended a five-match losing streak against seeded opponents at majors.
Isner had been 0-2 in third-round matches at the French Open.
No U.S. man had made it to the fourth round in Paris since Robby Ginepri four years ago.
Donald Young and Jack Sock can join Isner there with victories Saturday.
The last time three American men were in the fourth round at the clay-court major was 1995, when Andre Agassi, Michael Chang and Jim Courier got that far.