Falcao, who also became the first player to win successive Europa League finals with different sides having scored the winning goal for Porto last term, scored a 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 hot seat.
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 people 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."
Athletic Bilbao's coach Marcelo Bielsa took responsibility 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 responsibility increases in the sense that the difference in quality of the two teams is not the one shown tonight," added Bielsa, who coached his opposite number Simeone when he was in charge of the national side.
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