The British twins had broken free at the top of the final climb and it soon became clear there was little will from those behind to chase over the final couple of kilometres.
Two-time Tour winner Tadej Pogacar, also of UAE came, in third, 12 seconds adrift at the end of the 182km circuit starting and ending in the Basque city.
Pogacar raised his arms in triumph crossing the line, after seeing his newly-recruited teammate take the win.
"It was as if I had won the stage myself or in fact even better. It's a dream to see the team developing like this," said Pogacar, the 2021 and 2022 champion.
Yates took both the first yellow jersey for the overall lead, and the first green jersey for sprint points.
It was not the first 1-2 for cycling brothers as Andy Schleck beat his brother Frank on stage 18 of the 2011 Tour de France.
"Our parents are here too," Adam said after the stage. "I haven't seen them yet, they're somewhere along the coast in a campervan."
The younger by a few minutes, Adam led an elite clique up the Sondika hill through a narrow passage formed by enthusiastic flag-waving fans before allowing Simon to do the work most of the way to the line.
"He understood. I'm sure he'll get his chance to be a pain in the ass along the road somewhere," said the winner. "He asked me this morning what our plans were but I couldn't tell him."
Simon Yates, long considered the better of the identical twins, is second in the overall standings, at eight seconds, but appeared deflated after the finale.
"I've a fantastic relationship with my brother. I'm really happy for him," he said. "I'll stick it to him in a couple of days."
After the bonuses, Slovenian Pogacar is third overall at 18 seconds. Defending champion Jonas Vingegaard is another four seconds off the pace after tailing the Slovenian all the way to the line, as he did when he wore down his rival last year.
"The Tour de France won't be decided on four seconds," Vingegard said at the finish line.
There seemed to be little love lost as Pogacar asked Vingegaard without success to help break away on the final hill.
They did manage to create a good time gap, however.
The 2019 champion Egan Bernal, French former runner-up Romain Bardet, local hero Pello Bilbao and Australian hope Ben O'Connor all lost around 20 seconds to Pogacar and Vingegaard.
Basque colours eclipse Tour yellow
Saturday's race began under grey skies outside Athletic Bilbao's San Mames stadium before the riders were cheered around the countryside circuit by enthusiastic crowds.
Ecuadorian Richard Carapaz of EF team and Spaniard Enric Mas of the Movistar Team had hopes of contending for the final podium in Paris, instead both were forced out after being involved in the same crash towards the end of the stage.
Carapaz, the 2019 Giro d'Italia winner, finished the stage despite a bloodied left knee which was later confirmed to have "a small fracture" while Mas withdrew immediately after breaking his right shoulder.
The streets along the route were decked out with the Basque white, green and red flags as the Tour de France yellow was for once eclipsed.
Dane Vingegaard of Jumbo Visma led the peloton past the landmark Guggenheim museum before a breakaway formed as the peloton swept into pine and fir forests on the rolling green hills that mark this Atlantic coast region.
American rider Neilson Powless took the polka dot jersey for best climber after winning the race to the top of the penultimate hill.
Stage two is another hilly run through the Basque region from Vitoria to San Sebastian.
