The Colombian ace scored twice in the first half to send Los Rojiblancos on their way to a second Europa trophy in three years following their 2010 triumph over Fulham.
Falcao scored the only goal in the final last year to lead FC Porto to the Europa League and backed that up with another superb performance to finish the tournament on 12 goals from 15 matches.
Diego Simeone led his Atletico side to a first European championship in his debut season as manager on the continent, edging his former Argentine national team coach Marcelo Bielsa in the process.
Falcao produced a brilliant opening goal in the 7th minute when he advanced into the right side of the area before cruising past Fernando Amorebieta and sending a sublime curling left-footed shot into Gorka Iraizoz's top right corner.
Falcao made it 2-0 in the 34th minute when he received a neat cut-back from Arda Turan before managing to evade Amorebieta with a clever backheeled turn and then firing past Iraizoz again with his left.
Bilbao had a number of chances as they pressed forward in the second half but could not breach Thibaut Courtois' goal as Atletico managed to sneak a third on the break. Diego put in a lovely run, then came up with a nice stepover to help get past Amorebieta and Gaizka Toquero before hammering a beautiful left-footer across Iraizoz and into the back of the net to wrap up proceedings in the 85th minute and send the Atletico fans into raptures.