Chesterfield started the match strongly and Oliver Banks and Sam Morsy both saw shots saved in the opening five minutes.
Gnanduillet then tried to break the deadlock from long range but his shot was saved.
Carlisle finally broke the deadlock in the 75th minute when substitute Jabo Ibehre headed into the back of the net after Kevin Osei's cross.
Morsy tried to equalise immediately but missed the target, and Charlie Raglan's header was saved with seven minutes remaining.
Emmanuel Dieseruvwe forced the match to extra time in the 84th minute when he directed the ball into the bottom left corner.
The hero turned to villain though at the start of extra time after being shown a straight red card for serious foul play.
Osei looked to punish 10-man Chesterfield in the 102nd minute but his long-range effort was saved.
They were eventually punished in the 107th minute and, yet again, it was Ibehre who struck. This time, his shot found the bottom left corner.
Morsy missed the target with three minutes remaining and time ran out for Chesterfield.
Osei made sure of the result in stoppage time with Carlisle's third goal.