The game was finally balanced until the 35th minute when home defender Douglas Ramsey gave the ball away and McGeown was forced to bring down Dennis Wyness to prevent a certain goal.
McGeown was sent off and replaced by Eric Phillips with striker Jerome Vareille been sacrificed. Craig Dargo slammed home the resulting penalty and Caley were on their way.
Ten-man Ayr continued to battle although even their most ardent fans knew their hopes of pulling off a Cup shock were over.
Caley began to stretch the Ayr defence and it came as no surprise when they sealed the tie in the 76th minute.
Roy McBain floated over a cross and Wyness was left unmarked to score with a simple downward header.
Dargo chipped just wide as Inverness went close to adding a third and, while Ayr showed acts of endeavour, the damage had been done much earlier.