Cell 211 scooped a massive eight trophies at the ceremony in Madrid, including Best Film, Best Adapted Screenplay, Best Director for Daniel Monzon and Best Actor for Luis Tosar.
Agora, which stars Weisz as a female philosophy professor in Roman Egypt, picked up seven prizes with nods for Best Special Effects, Wardrobe and Best Original Screenplay.
Danny Boyle's Slumdog Millionaire was chosen as best European film.
Penelope Cruz missed out on the Best Actress statuette for her role in Pedro Almodovar's film Broken Embraces - the coveted prize went to Spanish actress Lola Duenas for Me, Too.
Almodovar - who quit the academy in 2005, citing disagreements with the voting procedures - received a standing ovation as he made a welcome return to the Goyas after a five-year absence, with a late appearance at the ceremony to present the Best Film prize.
And he credited academy president Alex de la Iglesia with facilitating his return, telling the crowd, "I'm here because you have a very persistent president. He pestered me until two days ago. And I had a lot of excuses, but he wouldn't take no for an answer."