The Colombian striker, top scorer in last season's Europa League with 17 goals as he led Porto to the title, netted the opener in the 23rd minute at a sun-bathed stadium after good work by former Arsenal winger Jose Antonio Reyes.
His second came from the penalty spot in the 36th after Brazilian Diego, another close-season addition to Gregorio Manzano's squad, had been felled in the box.
Atletico's third high-profile new signing, Turkey midfielder Arda Turan, then played Falcao through to complete his treble with a deft chip over goalkeeper Tono 10 minutes after the break.
Brought in to help fill the gap left by the departures of Diego Forlan and Sergio Aguero, the Colombian was given a rousing ovation by the home fans when he was substituted midway through the second half.
Turan claimed his second assist with just over 10 minutes left when he crossed for substitute Adrian to glance a header past Tono for the home side's fourth to round off their first win of the campaign.
Valencia made it three win in three thanks to a 1-0 win at Sporting Gijon on Saturday and are two points ahead of Barca, who drew 2-2 at Real Sociedad last weekend and 2-2 at home to AC Milan in the Champions League in midweek.
Real Zaragoza forward Luis Garcia had an eventful match against his former club Espanyol in Sunday's earlier kickoff, smashing in a stoppage-time winner for a 2-1 victory minutes after he seeing his penalty saved by Cristian Alvarez.
Garcia, who joined Zaragoza in the close-season, put his new club ahead in the 29th minute before Javi Lopez equalised with 20 minutes left.
In Sunday's Madrid derby, promoted Rayo Vallecano claimed their first win of the campaign with a 1-0 success at Getafe.
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.