Former world road champion Costa held off Lennard Kamna (Bora-Hansgrohe) and Santiago Buitrago (Bahrain Victorious) following the 158.5-kilometre ride from Pamplona to Lekunberri, as Evenepoel ended in fourth.
Costa won via a late tactical masterclass as he slowed down the pace and appeared to lull Buitrago and Kamna into a false sense of security before accelerating over the finish line to claim his first Grand Tour win since 2013.
"I believed in myself. The season started very well for me and now winning a stage of La Vuelta is very important for me and for the team. I'm very, very happy," said Portugal's Costa, who won the Volta a la Comunitat Valenciana in southern Spain in February.
German Kamna did well to finish second, overcoming a crash in the final three kilometres and getting back up on his bike to power past Colombian Buitrago.
"(Crashes like Kamna's) are things that happen in races," Costa said.
"We had already done that downhill once and I saw that he was going very fast so I hit the brakes because I knew that if we followed his pace, we could crash.
"It's a shame for him and then it was just about playing for the stage win. It went my way and I'm very happy."
Evenepoel, who claimed a superb victory on the mountainous 14th stage on Saturday after finishing 27 minutes off the pace on Friday's climbs, rode an aggressive race and made his intent to climb back up the rankings clear from the start.
The Belgian Soudal Quick-Step rider attacked several times throughout the race and engineered a breakaway with just over 80 kms left.
He ran out of steam in the final 15 kms and at one point was almost 30 seconds behind, but roared back with a desperate effort to finish two seconds slower than the leading trio.
Evenepoel is now just over 16 minutes off American Sepp Kuss, who retained the leader's red jersey. Kuss's Jumbo-Visma team mates Primoz Roglic and Jonas Vingegaard are second and third in the overall standings.
Monday is a rest day on the Vuelta. Tuesday's stage 16 is a 120.5-km ride from Liencres Playa to Bejes, which is mostly flat but ends with a steep climb.
