Quintana finished ahead of Chris Froome (Sky) and Esteban Chaves (Orica-BikeExchange) and Alberto Contador (Tinkoff) after a torrid battle in the final week of racing.
"I came into this Vuelta with a grudge after not being able to really contest the win in the Tour, yet very calm," Quintana said.
"The training rides I went on prior to the race showed that my fitness level was good, the data was telling me I could aspire to get a good result, and the team offered me their confidence from the start of the race.
"It kicked off with a great team time trial, where we lost by barely a fraction of a second, and from the very first week, my team-mates always kept me at the front and fought every single day so I didn't lose terrain on stages that didn't favour me.
"That was one of the keys to me winning this Vuelta."
Christian Knees (Sky) was the first rider to attack in the streets of Madrid.
The first real break took shape after Loic Chetout (Cofidis) jumped away from the pack, followed by Peter Kennaugh (Sky), Quentin Jauregui (AG2R) and Koen Bouwmann (LottoNL-Jumbo).
The fugitives built a one minute lead over the peloton led by Giant-Alpecin.
All the teams interested in a bunch sprint contributed to the chase and Chetout was the last rider to be reined in five kilometres from the line.
Orica-Bike Exchange went in the front of the bunch inside the final kilometre, perfectly setting up Nielsen for his second stage win in the race.
Daniele Bennati (Tinkoff) was second and Gianni Meersman (Etixx-QuickStep) finished third.
"This Vuelta win means a whole lot to me," Quintana said. "At the Tour, I reached the podium more out of class rather than legs.
"I didn't feel well in France, yet I found my best condition here. Also, it was a race with almost all all big GC names in the peloton present.
"A huge Chris Froome; Alberto Contador, who is one you must always keep an eye on; Chaves and Orica. Winning the race this way, and against them, makes it even more valuable."