The visitors played the best part of 50 minutes with only 10 men following Frankie Artus' first-half dismissal, but Martin Gritton's 18th-minute strike proved enough to earn a rare away win that fired the Spireites into the play-off places.
"Chesterfield defended with their lives but we didn't have enough guile or cuteness to break them down and that was a disappointment," said McIlroy.
"We had plenty of the ball and I couldn't have a go at any of the lads for their effort and commitment but in the final third we were very disappointing.
"After we came close on a couple of occasions I had the feeling that this wasn't going to be our day and that's how it turned out in the end."