Adam got the better of his identical twin to claim his first grand tour stage win, while his UAE Emirates team leader Tadej Pogacar finished third, also raising his arms in celebration.
Yates, who had set a devilish pace for Pogacar on the brutal climb shortly before the finish, pushed on in the descent to Bilbao, joined by his brother who is riding for the Jayco-AIUla team and already has two Tour de France stage wins to his name.
Adam was the stronger in the final uphill drag to prevail by four seconds, with Pogacar, who showed great legs in his first stage race since breaking a wrist in April, crossed the line 12 seconds off the pace.
"My brother came across to me and we started working together. At first, I didn't know if I should work with him, so I asked my team on the radio and they said 'Yeah, go for it'," said Yates, who wore the yellow jersey for four days in 2020.
"We're really close. To share this experience with him is really nice - but I wish he would pull a little bit easier because he almost dropped me one moment! I'm super happy, I just want to keep my feet on the ground."
It was the sixth one-two by brothers in the race's history after Andy Schleck beat Frank in 2011 while Henri and Francis Pelissier finished first and second four times between 1919-23.
Overall, Yates leads his brother by eight seconds after picking up 10 seconds for the stage win, while Pogacar is 18 seconds off the pace.
MAS AND CARAPAZ OUT
Pogacar took a four-second bonus for his third place to take a slight advantage over defending champion Jonas Vingegaard, who was the only main contender to follow the Slovenian when he attacked fending off a sea of Basque flags on the Cote de Pike 10 km from the finish.
"It's a team win, it's like I had won myself. That Adam seized the opportunity to get the yellow jersey is a team dream coming true," said Pogacar.
Pogacar had benefited from Yates's hard work to open a gap 500 metres from the top with only Vingegaard and France's Victor Lafay able to follow.
But the 2020 and 2021 Tour champion then slowed down, allowing a dozen riders to join them early in the descent, and eventually setting the stage for the Yates' brotherly battle.
Earlier, Spain's Enric Mas became the first of the 176 riders to abandon the race, quitting after crashing heavily on a descent some 22 kilometres from the line.
The Movistar leader, a three-time Vuelta a Espana runner-up who finished fifth overall on the Tour in 2020, fell off his bike alongside Ecuador's Richard Carapaz, one of the other podium contenders.
Carapaz of the EF Education-Easy Post team climbed back on his bike after being attended to by race doctors and was hovering more than 10 minutes behind the leading bunch, eventually rolling over the line a hefty 15:24 off the pace before pulling out of the race.
"Richard Carapaz will not start stage 2 of the Tour de France after an ultrasound scan revealed a small fracture in his left kneecap. He also needed three stitches to close the cut on the same knee," the team said in a statement.
