Kuss went into the penultimate stage of the race 17 seconds ahead of Jumbo-Visma teammate Jonas Vingegaard, with Primoz Roglic completing the team's domination of the general classification in third place, and the three crossed the finish line together.
Shortly before, Wout Poels outsprinted Remco Evenepoel to claim the stage win, after both riders were part of an early breakaway group.
With the Jumbo-Visma team happy to sit back and defend their top three places overall, a 31-man breakaway group on the opening climb led by last year's Vuelta winner Evenepoel was allowed to stay away and extend its gap.
The race went exactly to plan, as far as Jumbo-Visma were concerned.
"It was perfectly in control thanks to the guys in the beginning. Robert (Gesink) and Dylan (Van Baarle) were pulling for 90 per cent of the stage, it was a long hard day and they were out there all day, they were amazing," Kuss said after the race.
Stage 20 was the longest of this year's race, a 208-kilometre hilly ride from Manzanares El Real to Guadarrama. While none of the ascents were above category three, the riders did face 10 climbs before the final flat eight kilometres.
Evenepoel, whose general classification hopes evaporated on stage 13 where he lost more than 27 minutes on the leaders, seemed poised to win his fourth stage of this year's Vuelta.
The Belgian had three Soudal Quick-Step teammates alongside him for most of the race.
While Jumbo-Visma cruised along to safely guide the peloton, the breakaway was pushed by Quick-Step. They increased the pace in the final 50 kilometres, dropping riders along the way.
FINAL CLIMB
The final climb saw Poels attempt to get away from the rest and it looked like Evenepoel's race was over, but he rallied to rejoin the Dutchman along with three other riders.
But Poels, of the Bahrain Victorious team, put in a last-chance attack just before the final corner to get ahead of Evenepoel and held on to beat him in the final sprint, with Pelayo Sanchez finishing third.
More than 10 minutes later the Jumbo-Visma trio strolled to the finish line.
A beaming Kuss was flanked by this year's Tour de France winner Vingegaard and Giro d'Italia winner Roglic, with the team set to make history by becoming the first to win all three Grand Tours in one season - and with three different riders.
Kuss helped both riders to those victories, but in Spain, unexpectedly, it was the domestique's time to shine and he took over as leader after stage eight.
Vingegaard and Roglic both chipped away at Kuss's lead, but in the end, they guided the surprise leader to maintain his first place.
"Primoz did a lot of work for me there on the last climb, and then on the last flat with Jonas, that's just something I never imagined," Kuss said.
In the final moments of Saturday's stage, the three teammates put their arms around each other as Vingegaard and Roglic saluted their colleague.
"It was a super special moment to be able to relax that much in the last kilometre of a mountain stage with your two teammates is just really cool, it hasn't sunk in yet," Kuss added.
The Vuelta concludes on Sunday with a flat 101.5-kilometre ride from Hipodromo de la Zarzuela into Madrid.
