Garcia fired the Catalans ahead not long after the hour mark, before Stuani put the result beyond doubt late on as the Periquitos opened their account at the fifth time of asking.
The hosts started brightly but it took 16 minutes for them to threaten Getafe's goal, as Garcia saw his header expertly tipped away by Vicente Guaita.
However, Sergio Gonzalez's side failed to build on that early promise and the visitors nearly snuck ahead seven minutes after the interval.
It proved a decisive miss as Garcia opened the scoring 13 minutes later, collecting a Salva Sevilla pass before passing first time beyond Guaita.
Cosmin Contra's side looked keen to level matters but Getafe struggled to create the clear-cut chances necessary to do so.
Espanyol then took a big step toward their first league win this season as Stuani struck on 83 minutes, tucking home after Guaita saved from Garcia.