The Canadian, who was second in Monday's sixth stage, timed his kick superbly to open a gap to Omar Fraile with just 1,000 metres left of a gripping finish.
Astana rider Fraile had pulled clear late on as part of a cat-and-mouse chase for the line that also involved Alejandro Valverde, who finished third, and Nans Peters.
It was Woods who emerged triumphant after proving too strong for his rivals, as he claimed a second stage win at the Vuelta at the end of a somewhat messy race from Vitoria-Gasteiz in which he led a decisive breakaway approaching the summit of the final climb.
"That was a special day," said Woods, who won stage 17 two years ago. "I got away with those four guys, they ride super strong. I was able to sit in a bit, had a bit of luck, had the legs and managed to get the win.
"Initially, I wasn't supposed to get in the breakaway, however, the race just got so crazy. I'm going to savour this one."
Woods' EF Pro Cycling team-mate Hugh Carthy remains second in the General Classification, behind Richard Carapaz, who is also second in the points standings.
STAGE RESULT
1. Michael Woods (EF Pro Cycling) 03:48:16
2. Omar Fraile (Astana) +00:04
3. Alejandro Valverde (Movistar) same time
4. Nans Peters (AG2R La Mondiale) +00:08
5. Guillaume Martin (Cofidis) same time
CLASSIFICATION STANDINGS
General Classification
1. Richard Carapaz (INEOS Grenadiers) 28:23:51
2. Hugh Carthy (EF Pro Cycling) +00:18
3. Dan Martin (Israel Start-Up Nation) +00:20
Points Classification
1. Primoz Roglic (Jumbo-Visma) 79
2. Richard Carapaz (INEOS Grenadiers) 61
3. Dan Martin (Israel Start-Up Nation) 57
King of the Mountains
1. Guillaume Martin (Cofidis) 27
2. Sepp Kuss (Jumbo-Visma) 24
3. Tim Wellens (Lotto Soudal) 19
What's next?
Wednesday sees the riders depart Logrono, the Rioja region's capital, and set out on a 164km route through the mountains towards a gruelling final climb up to Alto de Moncalvillo.
The 2018 Vuelta winner Yates was in fifth place after the first 10 stages of this year's race but has had to pull out, just as he did in the Giro d'Italia earlier this year due to a knee injury.
Ninth-placed Ineos Grenadiers rider Pavel Sivakov was also required to withdraw after returning a positive coronavirus test.
Groves put a positive spin on the day for Yates' team, though, securing the win in Cabo de Gata after seeing off competition in a bunch sprint, finishing ahead of Danny van Poppel of Bora-Hansgrohe and Tim Merlier of Alpecin-Deceuninck.
"It feels fantastic," Groves said. "This morning with the news of Simon going positive for COVID-19, all the boys were disappointed. It's the best way to bounce back after such bad news."
The 23-year-old became the first BikeExchange rider to win a La Vuelta stage since Yates in 2018.
Elsewhere, world road champion Julian Alaphilippe's competition is over after he was taken to hospital with a suspected broken collarbone following a crash.
It leaves Alaphilippe's defence of his title at next month's World Championships in Wollongong in doubt.
Evenepoel maintains gap at the top
Remco Evenepoel is the first under-23 rider to lead La Vuelta for six or more stages since Dietrich Thurau in 1976 (six).
The gap to the lead remained unchanged as none of the frontrunners in the general classification troubled the stage leaders on Wednesday.
STAGE RESULT
1. Kaden Groves (BikeExchange-Jayco) 5:03:14
2. Danny van Poppel (Bora-Hansgrohe) same time
3. Tim Merlier (Alpecin-Deceuninck) same time
4. Juan Sebastian Molano (UAE Team Emirates) same time
5. Mads Pedersen (Trek-Segafredo) same time
CLASSIFICATION STANDINGS
General Classification
1. Remco Evenepoel (Quick-Step Alpha Vinyl) 39:39:04
2. Primoz Roglic (Jumbo-Visma) +2:41
3. Enric Mas (Movistar) +3:03
Points Classification
1. Mads Pedersen (Trek-Segafredo) 184
2. Remco Evenepoel (Quick-Step Alpha Vinyl) 85
3. Marc Soler (UAE Team Emirates) 81
King of the Mountains
1. Jay Vine (Alpecin-Deceuninck) 40
2. Robert Stannard (Alpecin-Deceuninck) 21
3. Jimmy Janssens (Alpecin-Deceuninck) 17
