Hincapie finished the 155-miles from Lezat-sur-Leze to Saint-Lary Soulan in the Pyrenees in just over 6 hours and 6 minutes. "I'm in shock right now," Hincapie said.
Lance Armstrong finished seventh but still retained the yellow jersey as overall leader. Italian Ivan Basso is in second place, 2 minutes and 46 seconds behind Armstrong.
Barring any setback, Armstrong is expected to win his seventh consecutive Tour de France title.
Sunday's stage brought the cyclists past the memorial of Fabio Casartelli, a teammate and friend of Armstrong who died on the course 10 years ago.
The Tour will take Monday off for rest.