The game was due to kick off at 9:30pm local 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."
PACKED CALENDAR
Real director of football Miguel Pardeza said the club did not want to risk another lighting failure and wanted to play the match on Monday at 5pm local time.
"We have a very packed calendar and Rayo obviously does not," Pardeza said.
"We have absolutely no responsibility for what has happened here and we have to defend our rights.
"As we have no guarantee that we can play tomorrow at 8pm, 5pm tomorrow is when we want to play."
However, Presa told Spanish TV that was "almost impossible" as many Rayo fans would not be able to attend.
The initial message he had received from the professional league (LFP) was that the game would be played at 8pm and Rayo would do everything to ensure the lights were back to normal, he added.
