The Colombia striker produced two moments of magic in the first half before Diego's late goal put the seal on Atletico's second title in three years.
Diego Simeone's side were vastly superior in their eagerly awaited all-Spanish final against Bilbao as Marcelo Bielsa's men froze on the big occasion.
Falcao's 11th and 12th goals of the tournament not only earned him the top scorer prize ahead of Schalke's Klaas Jan Huntelaar but took his all-time record in the competition to 29 in 29 games.
Bilbao had come into the tournament through the qualifying round, after they drew their home leg 0-0 with Trabzonspor but were spared a tricky return clash in Turkey after their opponents were promoted to the Champions League to replace Fenerbahce, withdrawn by the Turkish FA amid a match-fixing investigation.
They have thrilled and delighted onlookers since, never more so than in their win over Manchester United, but never truly got going tonight as their big names disappointed.
Arda Turan's cross provided the first Atletico chance for Adrian Lopez, who glanced his header across the face of goal.
Diego was unable to make clean contact after another high-tempo move but Atletico got their reward in the seventh minute.
Falcao bided his time on the ball in the right of the penalty area, worked a yard of space and curled a magnificent left-footed effort inside the far post.
Fernando Llorente headed Ander Herrera's free-kick wide, while being manhandled by Diego Godin, and was then too early on Markel Susaeta's cross and scuffed his volley wide.
Herrera was perhaps lucky only to be booked after hacking down Diego two-footed just outside the box.
Iker Muniain's long-range shot at the other end, having been gifted possession, was well saved by on-loan Chelsea goalkeeper Thibaut Courtois.
But Falcao doubled Atletico's lead in the 34th minute. Fernando Amorebieta dwelt terribly on the ball and lost possession on the edge of the box and, from Arda's cutback, Falcao brilliantly turned Jon Aurtenetxe and lashed home.
He then curled a right-foot effort high over the bar as half-time brought welcome respite for Bilbao.
Bielsa made a double-change at half-time, Ibai Gomez and Inigo Perez replacing Iturraspe and Aurtenetxe.
And his side started more brightly, Andoni Iraola playing in Muniain whose cross was cleared behind by Joao Miranda. Javi Martinez headed Ibai's corner over.
Diego fired just wide of Gorka Iraizoz's near post and a deflection robbed Gabi Fernandez's shot of its pace.
Perez's free-kick, swung in from 40 yards, almost caught out Courtois, who had to tip the ball around the post and Llorente was crowded out after failing to control the ball in the Atletico box.
Adrian curled wide before Amorebieta was booked for a nasty challenge on Falcao.
Miranda awkwardly diverted Ibai's cross behind but the corner came to nothing. Iraola's cross then narrowly eluded Llorente before Ibai hammered a follow-up volley just over.
Ibai brilliantly worked his way into the box but his cross was palmed away by Courtois and De Marcos spooned the rebound over.
Perez was booked for cynically chopping down Adrian and Gabi smashed a poor free-kick into the wall.
Susaeta's fierce left-foot drive deflected wide off Miranda and at the other end, Arda wasted some neat interplay with an overly audacious chip which Iraizoz easily fielded.
Coutois preserved Atletico's two-goal margin once more, saving from Susaeta after substitute Gaizka Toquero's shot was blocked by a defender.
Falcao danced past Javi Martinez but Amorebieta got back to deflect his shot on to the post.
Ibai took an air-shot at Toquero's cross before Diego put the game to bed.
The Brazilian took on Toquero and Amorebieta before firing low across Iraizoz. As Atletico celebrated, their opponents' despair was summed up by the sight of Muniain sprawled full-length on the ground in tears.
Ibai's long-range drive skimmed the crossbar in the closing seconds but there was no way back for Athletic.