Falcao came off the bench for Randy Nteka in the 71st minute and scored in the 82nd before saluting the ecstatic home fans at Vallecas.
The Colombian latched on to a through ball into the right channel from Pathe Ciss and fired a sizzling shot past David Soria and into the far corner.
Falcao has returned to Spain eight years after he joined Monaco from Atletico Madrid, where he became one of the world's deadliest strikers and won the Europa League, as well as the Copa del Rey.
But injuries have curtailed his form since and after scoring 19 goals in 34 league games for Galatasaray across two stop-start seasons, he was released, allowing Rayo to make their move.
Rayo are back in Spain's top flight after being promoted last term and have made an encouraging start, their win over Getafe giving them seven points from their first five games.