Falcao sparked the match into life in the seventh minute with a left-footed screamer from outside the Athletic box, and he added a second before half-time for what was his 12th goal in 15 Europa League matches with an empathic finish.
The prolific Colombian striker, who was denied a hat-trick by the woodwork, secured back-to-back Europa League successes after winning the tournament with FC Porto last season with Diego adding a third late on as the 2010 champions vanquished Marcelo Bielsa’s much-lauded side.
It was a stunning victory for Diego Simeone's side, with many of the Athletic players left to shed tears after a crushing defeat ended what has been an incredible European run.
Athletic, who made the rest of Europe take notice when they beat Manchester United home and away in the last 16, made one change from the semi-final with Oscar de Marcos back from a ban to replace Ibai Gomez, while Atletico called upon Gabi to slot in for the suspended Thiago.
Atletico, wearing their traditional red and white striped shirts with Athletic forced to sport their green change strip, stretched their opponents’ defence as early as the second minute as Adrian Lopez missed the target with a glancing header after beating his marker Jon Aurtenetxe at the near post.
Just five minutes later, Athletic’s defence was breached: Falcao twisted and turned on the right of the penalty area before unleashing a rasping left-footed strike into the far corner of the net with Gorka Iraizoz left flat-footed. It was a finish worthy of winning any final.
Athletic did enjoy fleeting forays of their own, and Fernando Llorente should have hit the target as he stole in front of Miranda to meet a precise cross from Ander Herrera, only to slice his volley wide of the near post from close range.
A tremendous strike from Iker Muniain from 30 yards forced Atletico goalkeeper Thibaut Courtois to hastily dive low to parry wide with Llorente unable to latch on to the rebound in the 24th minute, but it was all Atletico thereafter.
The irrepressible Falcao made his mark again in the 34th minute to double his side’s lead: the striker wriggled past his marker Aurtenetxe before firing a ferocious shot beyond the helpless Iraizoz from inside the Athletic box after Andoni Iraola had crucially dallied in possession. It was the forward’s 35th goal from 38 appearances this season.
Atletico continued to be lethal on the counter-attack as Athletic committed bodies forward and persistently left themselves horribly exposed at the back.
A clearly disgruntled Bielsa made two changes for the second half as Inigo Perez and Gomez were introduced to replace the ineffectual Ander Iturraspe and Aurtenetxe, and there was an instant response from his side.
Athletic were rampant at the start of the second half as they piled forward in their droves, but Miranda cleared the danger as Muniain got to the byline and almost created a clear-cut chance for Llorente.
Inigo Perez almost caught Courtois off guard with a very deep free kick from out on the right on the hour mark, and the goalkeeper was forced to frantically scurry back and tip the ball behind for a corner.
Bielsa made his final change minutes later as the frustrated Herrera was withdrawn with Gaizka Toquero introduced to a hero’s welcome, but Atletico continued to look assured at the back despite Athletic's tactical shifts.
Athletic continued to press forward incessantly late on, but clear-cut chances were scarce. Llorente knocked a high ball down with a header inside the Atletico box, but Gomez's shot zipped agonisingly over the crossbar with Courtois at full stretch.
The ebullient Gomez caused havoc in the final third with his mesmeric runs, and he wriggled past two challenges before cutting the ball back for De Marcos, but he lost his composure entirely as he fired high and wide from close range.
In what was a frenetic finale, Diego Godin and Miranda each made crucial last-ditch blocks to deny Athletic, while Falcao almost grabbed his hat-trick with a low drive rebounding off the near post with Iraizoz beaten.
But Diego extinguished any lingering hope from Athletic as he capped an incisive run with a clinical left-footed finish to beat Iraizoz at his far post and prompt jubilant celebrations from the Atletico players while a distraught Muniain collapsed in tears on halfway.
The closest Athletic, who only field players of Basque origin, had previously come was reaching the final of the UEFA Cup in 1977, where they lost to Juventus, and it was a second showpiece defeat for Bielsa’s side, while Atletico and the irresistible Falcao were left to celebrate yet again.