The hosts were unlucky not to double their advantage midway through the first half when Roberto Trashorras saw his curling effort from outside the box come off the post.
The Seville side themselves could have scored on the half-hour mark had Ruben Castro been able to keep his attempt from close range on target.
The second half saw both sides go at each other looking for the second goal of the evening, and it was Rayo who got it in the end.
Casano whipped in a cross inside the Betis penalty box that Delibasic headed home in the 64th minute.
Jose Carlos wrapped up the points for Rayo with seven minutes left to play, as he carved his way into the area and fired home into the top corner.