The game was due to kick off at 9:30 p.m. local time (20:30 British time) but the stadium was in semi-darkness and fans were still waiting outside the locked arena as workmen toiled to fix the affected lights.
Rayo president Raul Martin Presa said unidentified persons had cut the cables on some of the lights and when it became clear they could not be repaired in time the game was abandoned.
Discussions were underway to try to rearrange the match for Monday, Presa said at an impromptu news conference on the side of the pitch as some of the Real players took the chance to have a light training session.
"There were a huge number of fans waiting and we ask their forgiveness but we hope they understand that these are circumstances beyond the club's control," he said.
"They came here with all the excitement in the world hoping to see their team play one of the best sides in the world.
"Unfortunately due to an act of vandalism we are not going to be able to enjoy this match tonight."
Real initially said they did not want to risk another lighting failure and wanted to play the match on Monday at 1500, while Rayo insisted it should kick off at 1800 to allow more Rayo fans to attend.
"After the deplorable events that took place yesterday ... Real Madrid C.F. asked the Professional Football League (LFP) to schedule the game for Monday, during the day in order to avoid any potential problems that might again put it at risk," Real said in a statement on their website.
"The president of the LFP has decided, however, that the match will be played at 19:45 and has accepted personal responsibility for it to take place without incident," the statement added.
"Real Madrid C.F. bases their request on their concern for the lack of dates for the rescheduling of the match should it not be possible for the match to go ahead today."
Rayo said tickets for Sunday's match would be valid on Monday and fans who could not attend could return them.