Kuss is the first American to lead a Grand Tour since Chris Horner at LA Vuelta in 2013.
"It's nice and we'll see how it is over the next few days," Kuss said.
"Once you get the jersey you have to be a bit more in front and pay more attention to everything that's going on, rather than being able to relax in the peloton."
The peloton reeled in four breakaway riders at the start of the brutal final climb which pitched up to slopes of 20%.
Battle then commenced amongst the general classification contenders with Jumbo Visma's yellow-clad brigade again setting a fierce tempo along with the UAE Team Emirates trio of Juan Ayuso, Joao Almeida and Marc Soler and Belgium's defending champion Remco Evenepoel (Soudal Quick Step).
It proved too much for Martinez who fell away and Kuss was the first to attack before Evenepoel took it on over the summit on the downhill race to the finish line.
Evenepoel was unable to hold off three-time Vuelta winner Roglic in the final sprint to the line but looked strong as he came home in second place with Spain's Juan Ayuso third.
"I was on Roglic's wheel, I felt easy, I was following him but I didn’t know where the line was," Evenepoel said.
"I ended up losing some time to Primoz, which is a shame, and I feel a bit stupid actually."
Danish co-favourite Jonas Vingegaard, also of Jumbo Visma, was fifth with Kuss seventh.
With Frenchman Martinez more than a minute behind at the finish, Kuss now leads the standings by 43 seconds from Spain's Marc Soler (UAE Team Emirates) and one minute from Martinez.
Evenepoel, Roglic and Vingegaard are well-placed in sixth, seventh and eighth places respectively, around two and a half minutes behind Kuss but with two weeks of racing still to go.
Anyone assuming Kuss will just revert to his role of supporting more illustrious teammates may have to reconsider as the American again showed great legs.
"You always hope for the best. The guys did a really amazing job, controlling the really strong breakaway. Then we had no option in the end," Giro d'Italia winner Roglic said.
"In the sprint, it's always a bit of a gamble but I had the legs and I could do it. At least three (Jumbo-Visma leaders) now, no? But maybe there's still someone coming!"
More mountains are in store on Sunday with a 184-km route starting in Cartagena.