Cheltenham's third successive defeat has them peering anxiously over their shoulders after Grimsby won but there was no shortage of effort from the home team who hit the bar on three occasions.
Justin Richards hooked a shot against the top of the bar after 19 minutes while Matt Thornhill saw a thunderous left-foot shot rattle the woodwork just after the break.
Substitute Julian Alsop also headed a Josh Low cross against the bar as they pressed for an equaliser.
The Spireites posed the greater threat however, with Adam Rundle twice going close before the interval. Other chances fell to Simon Whaley who after the break forced two good saves from Robins keeper Scott Brown with long-range efforts.