Fuglsang, a silver medallist in the road race event at last year's Rio Olympics, went into the stage in third place.
But he managed to overhaul his rivals on a gruelling mountain ride, 115km long over three cols (Saisies, Aravis, Colombiere) before a final climb to the finish line at the Plateau de Solaison, high in the Alps.
The 32-year-old Dane soloed to the stage win, finishing 12sec ahead of Ireland's Dan Martin. Crucially, overnight leader Porte was 1min 15sec off the pace and Froome, who started the ride in second, a further 21sec adrift.
