Atletico captured its second title in three seasons, while Falcao won consecutive trophies with two different sides, having scored the only goal in the final for Porto last year.
He netted a spectacular opener in the seventh minute of this final and then added a second in the 34th. Diego completed Atletico's win in the 85th.
Falcao also topped the competition's scoring chart for the second straight year, and the the Colombia striker now has 29 goals in 29 games over two seasons in the Europa League.
Atletico coach Diego Simeone got the better of his former mentor Marcelo Bielsa. Simeone played for Argentina when Bielsa was the national team coach.
Atletico, which beat Fulham in the final two years ago, won its last 12 matches in this season's competition.
Atletico dominated the final before 52,347 mostly Spanish fans, who were hard to distinguish since both teams have red and white colors. Bilbao, being the "away" team for the occasion, wore green shirts.
Falcao struck his first after a good through ball from Diego. The Colombia striker was confronted by Fernando Amorebieta but the Bilbao defender slipped, lost a step and Falcao curled an unstoppable left-foot shot inside the far post in the upper right corner.
His second came after another error by Amorebiete, who carried the ball outside of his penalty box but carelessly lost it again to Mario Suarez, and Arda Turan collected it to set up Falcao. Falcao turned a defender and slotted home into the roof of the net from close range.
Amorebiete's miserable evening got even worse when he was easily rounded by Diego for the final goal.
Fernando Llorente had the first opportunity for Bilbao in the 29th, when he connected with a cross from Ander Herrera, but the striker did not have good control and put his shot wide.
The match was then briefly interrupted when two young fans ran onto the pitch but were hustled away by security personnel.
While Atletico controlled the early part of the match, Bilbao got back in after Llorente's opportunity and Iker Muniain had a shot blocked by Thibaut Courtois.
Diego missed narrowly early in the second half, while substitute Inigo Perez nearly caught Courtois off guard with a looping free kick from about 35 meters (yards) out.
Ibai Gomez, another Bilbao substitute, had a good effort go high in the 71st, as did Oscar de Marcos a few minutes later, with Courtois grounded.
A deflected shot by Markel Susaeta went close but wide of Atletico's goal as Bilbao sought to score and Susaeta again had a close-range shot blocked by Courtois.
On the other side, Atletico got more space to counterattack and Falcao hit the right post in the 80th.
In the last occasion of the match, Ibai Gomez struck the crossbar in injury time.
Only three other non-European coaches have won a major UEFA competition, and all three were Argentines: Luis Carniglia with Real Madrid, Helenio Herrera with Inter Milan and Alfredo di Stefano with Valencia.
Simeone became the third man to win the competition as coach and player. He played for the Inter Milan side that won the 1998 title, then known as the UEFA Cup. Dino Zoff won in 1977 with Juventus (beating Bilbao in the final) as a player and in 1990 as the coach. Huub Stevens won with PSV Eindhoven in 1978 and as coach of Schalke in 1997.