Brenner tailed leader Buitrago into the last 500 metres before launching his final attack, gritting his teeth as he sped ahead during the ascent to Sierra de La Pandera to become the first rider from his team to win a Grand Tour stage.
Red jersey holder Enric Mas (Movistar) of Spain did not challenge for the stage, a 154.5-km mountainous ride from Jaen, but finished 25 seconds ahead of Primoz Roglic (Red Bull–Bora–Hansgrohe), who dropped from second place to third overall.
Austrian Felix Gall (Decathlon CMA CGM) finished with Mas to take second place overall and trails by two minutes and six seconds in the race for the red jersey.
Cristian Rodriguez reduced his gap with compatriot Mas by more than four minutes as he finished third in the stage to move up to eighth place overall.
"I cannot really believe it yet," said 24-year-old Brenner. "I was struggling with some sickness earlier in the race and had a crash, but I really wanted to show myself and also my teammates... that I can be up there."
Mas in control
The peloton was nearly seven minutes behind when Kevin Vermaerke made a solo attack on the descent from Los Villares, with about 40 km left in the stage.
Mas ignored an early breakaway of more than 40 riders and stayed calm as he suffered a mechanical issue that made him briefly stop. Slovenian Roglic and Gall stayed close to the red jersey holder, who remained in the peloton till the final climb.
With 20 km left, Rodriguez joined a group of attackers to chase down Vermaerke. They caught the solo leader during the 12-km climb to Sierra de La Pandera, but in the final five kilometres Buitrago and Brenner sped ahead of the rest.
"I made the move, Santiago passed me, but I could catch up, and in the end I could beat him in the sprint... you live for days like today, I guess," said Brenner.
Mas attacked in the final kilometres, and Gall managed to stay on his wheel, but Roglic, who trailed Mas by a minute and 45 seconds coming into the stage, could not keep up, allowing the Spaniard to achieve an increased cushion of over two minutes.
The race continues on Sunday with a 181.2 km ride from Palma del Rio to Cordoba.
