Colombian striker Falcao struck two superb goals, having also hit the winner for Porto in the 2011 final before a big-money move to Spain, and Brazilian Diego snapped up a late third as Simeone triumphed over fellow Argentine Marcelo Bielsa.
Falcao, top scorer in the competition for a second straight season, netted a first-half double to put Atletico in control before Diego, who was suspended for the 2009 final when with Werder Bremen, sealed an emphatic win five minutes from time.
It was a fitting reward for Simeone who has revitalised 2010 Europa League winners Atletico since taking over in late December at the Calderon, where he became a fan favourite as a player in winning a league and cup double in 1996.
The 42-year-old former Argentina captain also lifted the UEFA Cup as a player with Inter Milan in 1998.
"Tonight I am feeling really strong emotions," Simeone told a news conference.
"A lot was at stake, people are very fond of me back in Madrid and they really expected this win... I'm really happy to give the fans of Atletico Madrid a new championship as a coach.
"We were very precise, very accurate... we felt comfortable on the pitch. Technically and tactically it was a good match for us."
Bilbao, playing in only their second European final and first for 35 years, tried to replicate the same attacking intensity that brought exciting knockout stage wins over Manchester United and Schalke 04.
But for the first 70 minutes Athletic, who only field players of Basque origin, had their cutting edge blunted by an Atletico side moulded in the steely ways of former midfield enforcer Simeone.
Imposing centre-forward Fernando Llorente and mohawk-haired teenager Iker Muniain have tormented defences all season but they got little change out of Atletico's well-organised rearguard, and it was not until the introduction of Ibai Gomez at halftime that Bilbao offered a threat going forward.
By then the impressive Falcao had almost sealed their fate.
The Colombian set a European competition record with 17 goals in Porto's triumphant 2010/11 campaign, including the winner in the 1-0 defeat of Braga in an all-Portuguese final.
Faced with having to replace the departed Sergio Aguero and Diego Forlan, the 40 million euros Atletico splashed out on Falcao appears to have been well spent.
He curled a delightful left-foot shot past flailing goalkeeper Gorka Iraizoz after seven minutes and bamboozled the Bilbao defence again in the 34th when he twisted and turned to fire home from close range.
It was a goal to savour, the front-man dragging the ball back to leave defender Jon Aurtenetxe on the floor before firing over helpless keeper Gorka Iraizoz who had already gone to ground.
Only the post denied Falcao a hat-trick 10 minutes from time when his deflected effort hit the left upright.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.