The 24-year-old joined the Saints from Gillingham in 2012 but has largely acted as the back-up goalkeeper, making just 23 appearances for the first team.
The Argentine will now spend the 2016-17 campaign in the Spanish second tier, with Southampton keen to provide the shot-stopper with regular first-team football.
"Paulo is a very popular player at the club and has developed steadily since his move from Gillingham, never letting us down when he has been called upon," executive director of football Les Reed told the club's official website.
"He is now at the point where regular competitive football is required for him to progress to the highest level. Playing regularly for Rayo Vallecano will provide invaluable experience for him. Rayo Vallecano is a progressive club, who are making a serious challenge to return to La Liga this coming season.
"Playing in Spain will also be valuable to Paulo in years to come when we hope he will be involved in regular European football. While Paulo has experienced junior football in his native Argentina and top-flight football in England, another culture and style will help him develop to his true potential."
Gazzaniga could make his competitive debut for Rayo when they take on Elche in the first match of the Segunda Division season on August 20.