Granada's Brazilian full-back Guilherme Siqueira netted twice from the spot after disputed handball decisions.
The Andalusians climbed to 17th with 17 points, ahead of 18th placed Real Mallorca on goal difference, and two ahead of bottom club Osasuna.
Getafe, who had led twice through Juan Rodriguez and Paco Alcacer, squandered numerous chances in the second half and failed to finish off their opponents.
Getafe took the lead when Rodriguez nudged in Adrian Colunga's freekick after 19 minutes, but the visitors levelled when they were awarded a penalty for a harsh handball decision against David Abraham, and Siqueira netted.
Striker Alcacer, on loan from Valencia, restored Getafe's lead when Colunga's clever overhead pass set him through on goal and he finished with a composed half-volley.
Getafe's Colunga hit the post on a breakaway, Alvaro Vazquez blasted a shot against the crossbar and then had a goal ruled out for offside which television replays showed should have stood.
At the end, a whipped in cross struck Getafe defender Rafa Lopez on the arm, and Siqueira sent Miguel Angel Moya the wrong way from the spot.