The reinvigorated Soudal-QuickStep rider crossed the line with arms aloft on the Larra-Belagua mountain finish, dropping Romain Bardet just over three kilometres from the end.
Last year's champion Evenepoel struggled badly on Friday explaining his world championships efforts back in August, where he won the time trial in Glasgow, had disrupted his training and scuppered his preparation.
The Belgian slunk over the line 27 minutes behind Jumbo-Visma rival and stage winner Jonas Vingegaard to end his hopes of competing in the general classification.
Sepp Kuss kept the red jersey with Jumbo-Visma still hogging the top three overall, with three-time Vuelta winner Primoz Roglic second, and back-to-back Tour de France winner Vingegaard third.
"Yesterday was a very difficult day, also the evening - I couldn't sleep too much, I had negative thoughts in my head," Evenepoel told Eurosport.
"Today I woke up and just told myself to make the best of it."
He added: "I think I can be very proud with this answer, after yesterday."
Stage 14 started with a relatively flat 50-kilometre stretch from Sauveterre-de-Bearn before climbing into the Pyrenees and back into Spain.
Evenepoel was on the attack from the very start in a blistering opening to the race on a baking hot day.
Eventually, he and 23 other riders managed to break away, with the peloton sitting back ahead of the Col Hourcere climb.
Evenepoel took maximum points in the race for King of the Mountain, driving on with DSM-Firmenich's Romain Bardet ahead of the break.
On the Puerto de Larrau, roared on by hoards of Basque fans, he clinched the blue-polka-dot jersey for best climber -- some consolation after his disastrous performance on Friday.
Evenepoel and Bardet hit the final 10 kilometres over four minutes ahead of the chase, ready to do battle for the stage, but the Belgian breezed away a long way before the finish.
"He's just way stronger, I was beaten by Remco," Bardet explained. "He's incredible."
The 23-year-old was emotional after his solo victory, his second stage win of this year's Vuelta.
The group containing the general classification favourites came in eight minutes 22 seconds behind Evenepoel, with no significant changes in the overall standings.
"(Evenepoel's win) is really impressive," said US rider and race leader Kuss.
"It's a nice way to bounce back after yesterday and I'm happy for him that he could get a win after yesterday maybe being a disappointing day - hats off to him."
On Sunday riders take on a hilly 15th stage, running 158.5 kilometres between Pamplona and Lekunberri.
This is the 78th edition of the Vuelta, which ends in Madrid on September 17 after 21 stages and 3,153.8 kilometres.
