The visitors took the lead after only a minute when Falcao converted from the penalty spot following Vadim Demidov's handball.
Adrian Lopez had a great opportunity to double his side's lead on 23 minutes, but he missed the ball from six yards out.
Atletico did stretch their advantage a minute into the second half as Lopez slotted in after his free header had been saved by Claudio Munoz.
The game was wrapped up in style in the final 10 minutes as Falcao chipped the goalkeeper from outside of the area after he had latched on to the through ball.
The striker completed his hat-trick in injury time when he arrived in the box to sidefoot in Koke's cross.
The win has moved Diego Simeone's side up to seventh in the La Liga table, four points off the Champions League places.