Preston were 2-1 up entering added time at the end of the game when Carl Baker leveled for the home side, before Clarke stuck.
Clarke, who is still the top scorer at Glanford Park this season, walked the ball into the net to notch his first for the club and to send the Sky Blues through.
“The finish to the game was unbelievable,” said Robins.
“Hopefully that first goal will set Leon off on a run because he has straight away endeared himself to the support. And me".
A dejected Preston boss Graham Westley suggested Clarke was offside.
“It’s always tough to accept a decision that is as blatant as that but we didn’t get the rub of the green with the decisions all night," he said.