Tour de France champion Wiggins attacked out of an elite group inside the final kilometre but Movistar's diminutive Colombian climber Quintana produced a devastating turn of speed to take the victory.
His team-mate Alejandro Valverde was second over the line and takes the overall race lead in the seven stage World Tour event from Gianni Meersman, winner of the first two stages.
Martin Kohler (BMC), Lucas Haedo (Canondale), Karol Domagalski (Caja Rural) and Nicolas Edet (Cofidis) were involved in the break of the day, attacking shortly into the 180km route from Vidreres.
They built up a lead of 7:30 after 20km and maintained that advantage with 80km to go following the category one climb of Alt de Sant Hilari.
Movistar led the peloton and reduced the gap as Edet and Domagalski left their fellow escapees on the cat 1 ascent of Tunel Collabos.
Edet went solo on the 12km concluding climb to the summit finish but was caught and passed by Jurgen Van Den Broeck (Lotto-Belisol) with 4km remaining.
But the Belgian was reeled in by the Sky train before their team leader Wiggins made his move for a first victory since the Olympic time trial.
However 23-year-old Quintana, who had seven victories in his debut European season in 2012, had other ideas and crossed the line ahead of Valverde and local favourite Joaquim Rodríguez.
Wiggins was fourth and is third on GC, 10 seconds behind 2009 Vuelta champion Valverde and four behind Katusha's Rodríguez.