In what eventually turned out to be a first half of few concrete chances, Tiago could have put the home side in the lead as early as the third minute with a header from close range, but his effort went wide.
With eight minutes left until the half-time interval, Alberto Lopo inadvertently put the ball inside his own net to double Atletico's advantage.
Things went from bad to worse for the visitors when they were reduced to 10 men on 42 minutes as Valera was sent off for two bookable offences.
The floodgates opened after the break as Atletico began to take advantage of their numerical superiority.
Garcia could have completed his hat-trick just before the hour mark had his effort from close range not hit the Getafe crossbar.
Adrian wrapped up the proceedings in the third minute of injury time when Villa teed him up for an easy finish from the centre of the Getafe box.