Wayne Curtis nearly had the Shrimps level in the 18th minute but his volley came back off the post
Sammy McIlroy's side failed to take advantage of Chesterfield having a man dismissed
in the first half when Frankie Artus was sent off for a second booking.
The Spireites were denied a second when Barry Conlon's overhead kick hit the post but the home side struck the upright twice more through Phil Jevons and Craig Stanley.
Tommy Lees saved well from a Stanley header and a last gasp Will Haining strike but the Chesterfield defence held out.