In the 28th minute, Tito and Leonel Galeano both went close with headers from corners, before Youssef El-Arabi hit his shot against the upright from fewer than six yards following a quick Granada breakaway.
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The away side took the lead in fortuitous circumstances in the 35th minute, as Rico's free kick from the left touchline evaded everyone in the box and nestled in the bottom corner.
The referee was incredibly busy in the opening period, with no fewer than six yellow cards being handed out before the break.
Nacho picked up two bookings in the space of six minutes to leave his trailing side with 10 men for more than half of the game.
The 10 men of Rayo did not let the dismissal affect them, though, and Jonathan Viera saw his shot pushed away by Roberto three minutes into the second half.
Bueno was denied twice in quick succession by a busy Roberto as the home side peppered Granada's goal in an attempt to get back into the game.
Granada almost put the game beyond doubt in the 75th minute, but Riki sliced his effort wide from 12 yards.
However, the Spaniard wrapped up all three points with a minute left to play when he tapped home Yacine Brahimi's pass from close range.