The visitors, who compete in the third tier of Spanish football, took a shock lead at the Bernabeu 25 minutes into the game.
The 20-year-old forward then edged the hosts in front on 70 minutes, slotting past Pol Freixanet on the rebound after the goalkeeper kept out Bale's initial effort.
However, the Segunda Division B side went on to claim a draw on the night through Alvaro Portilla, who rattled home on the second attempt with just one minute left to play.
Real Madrid (4-3-3): Navas; Tejero, Achraf, Nacho, Theo; Llorente, Ceballos, Kovacic (Seoane 81'); Feuillassier (Bale 62'), Mayoral, Oscar (Quezada 75')
Fuenlabrada (4-3-3): Pol, Athuman, Diaz, Marrero, Garcia; Cristobal, Milla, Bravo (Portilla 69'); Annor (Quero 79'), Barrozo (Villalba 61'), Fraile