It was Granada who had the first chance of the match when Diego Buonanotte found Yacine Brahimi inside the box, but the Algerian's volley was high and wide.
Things took a turn for the worse for the hosts just after the half-hour mark, however, as captain Pape Diakhate was shown a second yellow card for an elbow on Diego Castro.
Getafe almost took the lead right on the stroke of half time when Castro threaded the ball through to Pablo Sarabia, but the Spaniard could only put his shot wide of the post.
There was more success for the visitors immediately after the break, however, as Pedro Leon's cross was turned into his own net by an unfortunate Jeison Murillo.
Leon was involved again less than 10 minutes later as he smashed a powerful effort into the top corner to double Getafe's lead.
Granada came close to pulling themselves back into the match with just under 15 minutes remaining, but Piti could not make enough contact on Michael Pereira's cross to trouble the keeper.
Getafe almost put the icing on the cake in second half stoppage time when Ciprian Marica fired towards goal, but his effort came back off the crossbar as the visitors were forced to make do with a 2-0 win.