The result left Atletico in third place, two points behind Barcelona and five behind league leaders Real Madrid -- who both have a game in hand -- while Rayo remain in 10th.
Atletico's hard pressing yielded the opening goal in the 20th minute when Antoine Griezmann pinched the ball in Rayo's defensive third and found Alvaro Morata in the box for the Spaniard to tuck away his fifth goal of the season.
Falcao was often the focus of the visitors' attack but the 36-year-old did not get clear-cut chances to score while Atletico loanee Sergio Camello also went close to scoring against his parent club with a shot that went just wide.
But Rayo received a lifeline in the 90th minute when the referee awarded them a penalty for Jose Gimenez's handball following a quick VAR check.
Falcao, who scored 70 goals for Atletico between 2011-2013, stepped up and scored with aplomb and though the Colombian striker refused to celebrate, his teammates engulfed him as they came away from the Metropolitano with a precious point.