Woods excelled on the route from Lugo to Puerto de Ancares, securing his fourth Grand Tour stage win and his third in La Vuelta.
The leading group of five riders remained tightly packed as they approached the final, gruelling ascent. However, Woods broke free 4.5 kilometres from the finish line.
Switzerland's Mauro Schmid finished second, trying but unable to match Woods's pace in the mountains, while the local audience cheered as Spaniard Marc Soler claimed third place.
"It's been a tough race. And this is a big moment of catharsis, a big release. The pressure has been building and to get this win now, I’m on cloud nine," Woods said.
"I knew that as long as I had the guys within striking distance I had a shot at winning because the climb really suited me.
"I didn’t want to go as early as Mauro (Schmid) did but then I had to. It was a long four kilometres to be alone."
The Canadian pointed to the flag on his jersey as he crossed the finish line, celebrating his hard-fought and strategic win.
Ben O'Connor's lead in the overall standings was slashed by almost two minutes by Slovenia's Primoz Roglic. The Australian now leads by just one minute and 21 seconds.
"I was pretty cooked. Sad times but I’m still in (the leader's) red (jersey) so at least it’s a good thing," said O'Connor, who finished 33rd in the stage, compared to Roglic's 16th.
"Tomorrow, I’ll just try to do my best to recover, have another day in the red jersey and see how Sunday goes."
