Falcao, known as 'El Tigre', scored from the penalty spot, with a superb lob and wrapped up the scoring to earn Diego Simeone's team the three points in an impressive display in San Sebastian.
The result left Atletico on the brink of the European places in La Liga in seventh with 26 points from 19 games.
Earlier, Espanyol climbed to fifth on 28 points with a 3-0 home win over Granada thanks to goals from Raul Baena, Joan Verdu from the penalty spot and Rui Fonte.
They were two points ahead of Bilbao, Atletico and Sevilla, who held on for a 1-1 draw in the city derby at Real Betis despite playing the last 20 minutes with 10 men.
Elsewhere, Getafe came from behind to beat struggling Racing Santander 2-1 away.
Atletico striker Falcao opened the scoring from the penalty spot after only two minutes at a rain-soaked Anoeta and Spanish youth international Adrian doubled the lead early in the second half after a well-worked move.
Falcao saved his best until the 82nd minute, a lob from outside the area leaving home goalkeeper Claudio Bravo stranded, and he notched his 14th of the campaign at the end from Koke's pull-back.
The first Seville derby in three years looked to be heading the way of promoted Betis when midfielder Benat curled a freekick over the wall in the 26th minute.
Sevilla drew level just before the break when striker Alvaro Negredo headed in from a Jesus Navas cross, and although they lost defender Federico Fazio to a harsh second yellow card in the 70th minute they held firm.
The result failed to lift the pressure on under-fire Sevilla coach Marcelino, however, as they extended their winless run to five games.
