Yates was part of an early breakaway group but, with 58 kilometres still to race, the UAE Team Emirates rider went off alone and held off the challenge of Ecuador's Richard Carapaz (EF Education-EasyPost) who finished one minute 39 seconds behind.
O'Connor outsprinted Mikel Landa and Florian Lipowitz on the line to take third on the stage and claimed four bonus seconds to increase his lead over three-times former champion Primoz Roglic (Red Bull-Bora-Hansgrohe) to three minutes 53 seconds.
Carapaz moved up to third overall and Yates was rewarded for his efforts when he went from 27th to seventh in the general classification after his first Vuelta stage win.
The 178.5-km ride from Motril to Granada was the toughest stage of this year's race so far, with three category-one climbs and the riders tackling the Alto de Hazallanas twice before descending into Granada for the finish.
Wout van Aert was the first to launch an attack from the start and eventually a group of 26 riders formed, but Yates and his UAE teammates Marc Soler and Jay Vine pushed the pace, dropping riders on the first climb.
Carapaz attacked from the peloton and went on a solo chase, while the lead group splintered further with just three riders remaining as they went up Alto de Hazallanas for the first time before Yates went alone.
Yates managed to stretch his lead while, behind in the peloton, Roglic's team upped the pace to split the main bunch, but O'Connor (Decathlon-AG2R La Mondiale), who had seen almost a minute shaved off his lead the previous day, held firm.
Spain's Enric Mas (Movistar) attempted to get away, looking to protect his third place overall, but he was caught with one kilometre to the finish and O'Connor showed he had plenty left in the tank as he took third place, with Roglic in eight.
After Monday's rest day, stage 10 on Tuesday will be a 160-km ride from Ponteareas to Baiona, another mountain stage.