[1][2] Although she was not out while competing, Rodríguez became the second competitor in a major international beauty pageant to come out as LGBT, following Julia Lemigova, who represented the Soviet Union at Miss Universe 1991.

Rodríguez was selected as one of the twelve delegates at the inaugural Miss Universe Spain 2013 competition, held in Madrid.

Rodríguez represented Spain at Miss Universe 2013 on 9 November 2013, finishing as 1st runner-up to María Gabriela Isler of Venezuela, her roommate during the competition.

She is also only one of seven former Miss World semifinalists to place in the Miss Universe semifinals, the others being Michelle McLean of Namibia in 1992, Christine Straw of Jamaica in 2004, Ada de la Cruz of the Dominican Republic in 2009, Yendi Phillips also of Jamaica in 2010, Catriona Gray of the Philippines in 2018, Julia Gama of Brazil in 2020, and Andrea Meza of Mexico in 2020 (coincidentally Rodriguez, de la Cruz, Phillips, and Gama all finished as 1st Runner-Up).

This revelation sparked international headlines referring to Rodríguez as a lesbian, with Rodríguez becoming the first major national pageant titleholder to be out as LGBT, and the second woman out as LGBT to have competed in one of the major international beauty pageants, following Julia Lemigova, who represented the Soviet Union at Miss Universe 1991.