Jumbo-Visma rider Roglic extended his lead in the overall standings to two minutes 48 seconds after his closest rival, Spain’s Alejandro Valverde, lost 23 seconds on the Slovenian in the spectacular Asturian mountains.
The Danish rider’s chances in the Tour de France were ruined by two crashes with Astana saying he was in the form of his career. The 34-year-old emerged from the 21-rider breakaway to leave his opponents behind and claim a lone triumph.
British Team Ineos cyclist Tao Geoghegan Hart came in second, 22 seconds behind Fuglsang, after finishing third in Stage 15 on Sunday. Fuglsang's team mate Luis Leon Sanchez, took third ahead of Deceuninck-QuickStep Briton James Knox.
“It feels great,” said Fuglsang. “It was one of my goals to come here but it’s one thing to have a goal and another to fulfill it. I’m super happy to have this win.”
Riders attacked from the start in a constant battle for the break at the start of the 144.4km route between Pravia and Alto de la Cubilla.
Roglic and his rivals were several minutes behind the breakaway group, with the Slovenian reacting to several attacks from Miguel Angel Lopez and Tadej Pogacar.
Movistar rider Valverde struggled and Pogacar gained on him, with the Slovenian now 54 seconds behind in the Spaniard in the general classification.
“It was a hard day with a long last climb," Roglic said. "And a good day for us in the end. [Lopez] attacked really really strong. I was looking at Valverde and when there was a gap I didn’t jump. When I saw Valverde was having a bit of difficulty I went and I bridged the gap.
Stage 17 takes place on Wednesday after a rest day. This stage is marked as flat in the guidebook and is the longest so far at this year's Vuelta at 219.6km from Aranda del Duero and Guadalajara.
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