The left-back scored twice from the spot to ensure his side climbed out of the relegation places, but the Azulblancos will be aggrieved to have squandered the lead twice
Juan Rodriguez put the Madrid outfit ahead early in the first half, before Guilherme Siqueira scored a penalty to level the scores soon after, but when Paco Alcacer appeared to have sealed the win, the Brazilian scored a second to secure a share of the spoils late on.
Despite sitting 11th in the table, Luis Garcia's Getafe side were stuck in a rut, having failed to win any of their last six matches. Granada, meanwhile, were living dangerously themselves as a run of two wins in 12 had seen them drop into the bottom three.
Inside 18 minutes, Adrian Colunga's free kick was guided in by former Deportivo midfielder Juan Rodriguez to hand Los Azulblancos the lead at the Coliseum Alfonso Perez.
However, moments later, Abraham handled inside the box, allowing Siqueira to drive the ball into the bottom left corner of the net from 12 yards out and restore equilibrium in the contest.
Nonetheless, on the cusp of half-time, Colunga played the role of provider again, feeding Valencia loanee Alcacer, who duly applied the finish to place Getafe back in the driving seat.
The match remained on a knife edge until stoppage time when Rafa was penalised for another handball offence, and Siqueira came up trumps from the penalty spot again.
The result means Granada climb out of the relegation zone at Mallorca's expense, but Getafe, despite throwing two points away, consolidate their position in mid-table.