Falcao, who also became the first player to win successive Europa League/UEFA Cups with different sides having scored the winning goal for Porto last term, scored a brilliant first-half double to take his tally for this season's competition to 12.
Brazilian attacking midfielder Diego, who missed the 2009 final for Werder Bremen against winners Shakhtar Donetsk because he was suspended, added a third late in the second-half to give Argentinian coach Diego Simeone a trophy just months after he took over the hotseat.
Atletico's victory also saw them break Barcelona's record of successive wins in European competition as they recorded their 12th to break the record of 11 set in the 2002/03 Champions League campaign.
Falcao, named after the great Brazilian footballer of the 1982 and 1986 World Cup teams, said he was especially pleased because peopel had advised him not to go to Atletico.
"I cannot contain my joy," said the 26-year-old.
"After all that has been said, like that I made a mistake in going to Atletico, I showed them it was no mistake.
"We had highs and lows but we deserved to finish as we have done. The first goal? I didn't have an angle so I simply looked for some space.
"They told me to look for the far post from time to time, and this time I listened," he said with a smile.
"As for my future well we don't know whether Atletico will play in the Champions League or Europa League next season.
"We still have a possibility of the Champions League and we are determined to achieve that."
Simeone, who was a hard tackling midfielder who won over 100 caps, paid tribute to his players.
"Winning is always marvellous, but you enjoy it more as a player, because one is on the pitch, one can scream, run, do a lap of honour," said the 42-year-old.
"When you are a coach you leave that sort of thing to the players. I am proud of the players, happy for the fans, and thank those who brought me to the club in December."
Athletic Bilbao's coach Marcelo Bielsa took responsiblity for the defeat.
"I am disappointed, we did not play a good match," said the former Argentina coach, who guided the Argentinians to the 2004 Olympic title.
"I am in charge of the strategy of the squad who failed to achieve its objective, which makes the disappointment even greater.
"Atletico deserved their win, but the scale of it was excessive.
"The feeling of personal responsiblity increases in the sense that the difference in quality of the two teams is not the one shown tonight," added Bielsa, who coached Simeone when he was in charge of the national side.
Falcao opened the scoring on seven minutes with a spectacular goal taking the ball down the right and then into the box and as defender Fernando Amorebieta slipped he curled the ball round three defenders and the keeper from the edge of the box for his 11th goal in 15 Europa League games this season.
Bilbao's far younger side - an average age of under 24 - took a while to settle but gradually started to make more of an impact.
Their leading striker Fernando Llorente - scorer of 29 goals this term - had missed with an early header when he was not helped by being shoved in the back but he should have done better in the 19th minute.
Found brilliantly in the box he only had Belgian youngster Thibault Courtois to beat but his usual sure touch let him down and his shot skimmed off his boot and wide.
Falcao, though, was unstoppable at the other end and his second stemmed from the hapless Amorebieta being dispossessed as he tried foolishly to dribble it away from the box.
Turkish playmaker Ardi Turan found the Colombian, who completely fooled centreback Javi Martinez sending him the wrong way as he went the other before curling the ball into the top far left corner to double Atletico's lead.
The two sides were more evenly matched in the opening 15 minutes of the second-half with Diego going close for Atletico but finding the side netting while Inigo Perez forced a good tip over the bar from 19-year-old Courtois.
Athletic kept pressing but clear chances were rare, though Gomez Ibai had some Atletico fans hearts beating faster with his volley from outside the area which just cleared the bar with 20 minutes remaining.
Markel Susaseta went even closer as the ball ping-ponged round the Atletico area and the ball fell to him unmarked five yards out from goal but with just Courtois to beat the 'keeper - on loan from Chelsea - produced a great block with his legs.
Diego, though, sealed the win for Atletico with a fine solo goal with five minutes remaining, breaking through a weak tackle by the demoralised Amorebieta and firing home from the edge of the area.