The veteran Portuguese Intermarche–Circus–Wanty rider beat out Lennard Kamna and Santiago Buitrago in the final stages to cross the line at Lekunberri first by a fraction.
Remco Evenepoel, whose Grand Tour defence crumbled on Friday, enjoyed another strong ride after Saturday's stage win, but ran out of steam ahead of the denouement.
Sepp Kuss kept the red jersey, sitting ahead of his Jumbo-Visma team-mates Primoz Roglic and Jonas Vingegaard in the top three of the general classification.
"I believed in myself from the first moment," said Costa, 36, celebrating his fourth Grand Tour stage win and his first for a decade.
"A victory in the Vuelta is very, very important for me and for the team, I am so happy."
A relatively flat start to the race encouraged attacks, with Evenepoel looking determined again, but they were all reeled in, with the peloton intact for the first 50 kilometres from Pamplona.
On the first climb of the 158.5 kilometre run, Puerto de Lizarraga, Evenepoel attacked again, with Buitrago and Juan Pedro Lopez keeping up.
Several more riders followed and that move was curtailed too, with Jumbo-Visma driving the peloton on, not keen on allowing Marc Soler or Aleksandr Vlasov to get ahead of them.
Evenepoel, in blue and white as the best climber, launched a new attack and added to his King of the Mountain points tally at the top of the climb.
The Belgian, Kamna, Buitrago and 12 others established the day's breakaway, this time with Jumbo-Visma content to allow them to stretch the gap a little.
On the second climb, category two Puerto de Zuarrarrate, Buitrago made a move but Evenepoel brought him back.
The reigning champion took maximum climbing points again at the summit and, setting a fine pace, forced several riders to drop.
On the second and final ascent of Puerto de Zurrarrate Buitrago and Rui Costa eased ahead of Evenepoel and the rest of the chase group.
After doing a lot of work earlier in the stage - not to mention Friday's heroics - the Soudal-QuickStep rider could not stay in contention for the stage win.
Kamna, who won stage nine, kicked on to overtake the leaders but slid off the road into a ditch. He recovered to catch up with the flagging Buitrago and the cunning Costa.
Costa sprinted over the line just before Kamna to conclude an enthralling tactical tussle between the three riders.
Riders will enjoy the race's second rest day on Monday before Tuesday's stage 16 run to Bejes from Liencres beach, where an explosive uphill finale is anticipated.
