The hosts secured the three points thanks to goals from Jon Daly and Fraser Aird before Bilel Mohsni put the match beyond Ayr with five minutes to go.
Manager Ally McCoist said: "The first half was fairly comfortable and we scored a couple of good goals.
"I just felt we didn't raise the tempo of the game in the second half at all. It stopped us from killing off the game until later on when we got the third goal.
"But it's difficult to be overly critical because it's another great result.
"There's always the potential for slip-ups but we will do our best to make sure complacency doesn't set in."
There was a pre-planned protest organised by fans group Sons of Struth during the game but McCoist kept his counsel.
"I said yesterday that the fans, who have been fantastic, would give the team their continued support. We feel they are doing that," he added.
"What they've got to say, and how they feel, they are well within their rights
to do that but I'd rather not comment on it at all."
Ayr United boss Mark Roberts added: "The guys maybe gave them too much respect in the first half.
"We had a few teething problems but we got to grips with the game in the
second half and we passed the ball better and had a bit more belief.
"I think the third goal was a bit harsh on us but Rangers are flying and will probably win most of their games.We were just disappointed with the three goals."