It had previously appeared as if the hosts would hold out to halt their recent losing streak, but the former Manchester United striker struck on 94 minutes to condemn them to a fourth consecutive defeat.
The two sides were level in the table on seven points prior to kickoff, but Rayo dominated the first half without creating many clear-cut opportunities, as Lica's high shot was saved by Granada keeper Roberto.
A 20-yard Roberto Trashorras strike which narrowly missed to the left was as close as they came to breaking the deadlock, while Granada's Jeison Murillo headed Fran Rico's set piece over on 39 minutes.
Granada started the second half the brighter but strong defending from Vallecano kept them out, and the visitors then regained control of the game around the hour mark as Lica and Emiliano Insua had decent efforts.
The action calmed again as the two teams appeared content to play out a draw, but there was a late twist as Chelsea loanee Gael Kakuta found Manucho to fire into the bottom corner deep into stoppage time.
The result takes Rayo into the top half of the table, while Granada remain 13th and winless in October.