The American took the stage win after he attacked on the last of the four first category climbs with six kilometres remaining of the punishing 154.4km mountain stage from Tineo to Santuario del Acebo in Asturias to catch up with lone leader Sergio Samitier (Euskadi-Murias).
Kuss, 24, held his own on the summit finish, coming over the line 39 seconds ahead of second-placed Ruben Guerreiro (Astana) to earn his first Grand Tour stage win, while Tao Geoghegan Hart of Team Ineos came third, 40 seconds behind.
“I wanted to ride within myself and have a little bit in reserve in case Primoz came across. Maybe it was a bit too early to go, but I felt good,” Kuss said.
“I knew that the first few kilometres of the climb were the hardest so I tried to be in a good position and wanted to ride at my own rhythm. There were no mind games.”
Valverde (Movistar) launched an attack on Roglic shortly afterwards in a bid to reduce his overall deficit although the race leader was up for the fight and the pair stayed together to come eighth and ninth respectively, two minutes 14 seconds down on Kuss.
Valverde still trails Roglic by two minutes 25 seconds but the Vuelta now looks set to be a two-horse race after the pair caused considerable damage to the other contenders.
2016 winner Nairo Quintana (Movistar) suffered the most, finishing three minutes 50 seconds behind Kuss and now trails Roglic by five minutes nine seconds.
His compatriot Miguel Angel Lopez (Astana) also saw his hopes of an overall win fade as he lost 41 seconds on Roglic.
Lopez came over the finish line at the same time as Tadej Pogacar (UAE Team Emirates), who remains third overall but fell three minutes 42 seconds behind Roglic and lost 40 seconds to Valverde.
“My legs were feeling good so I decided to attack from afar and I have to say thanks to Roglic and to (Movistar team-mate) Marc Soler for working so hard,” Valverde said.
“After all that hard work it would have been very bad if I hadn’t gained anything but 40 seconds is a very good amount.”
The Vuelta continues tonight with Stage 16 stretching 144.4km from Pravia to Alto de La Cubilla. Lena, another punishing mountain stage LIVE streaming to Cycling Central / SBS OnDemand from 9.15pm AEST and broadcast live on SBS Viceland from 10.40pm AEST.
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