The result keeps the Robins three points above the relegation zone with four games to play, while Chesterfield missed out on an opportunity to close the gap.
Both sides displayed visible nerves in a game of few chances that was stop-start in nature, with both sets of midfield players battling to gain an advantage.
The best chances of the opening 45 minutes fell to the visitors with small, nippy forward Jamie Ward looking the most likely source of a goal.
Ward turned sharply on to an Alex Bailey pass in the box, but dragged his shot across the face of goal then, in first-half injury-time, the former Aston Villa trainee sprinted forward and shot just wide of goal having collected a loose pass from Town's David Bird.
The best chance, however, arrived after 20 minutes when Paul Shaw headed down a corner kick from Gareth Davies at the far post and Hungarian defender Janos Kovacs spun and shot over the crossbar from less than six yards out.
Cheltenham's best chance came three minutes after the break when midfielder Bird let go a fearsome volley from the edge of the box that goalkeeper Barry Roche did well to parry.
The Robins applied heavy pressure to the visitors' goal at times in the second half, but the Spireites backline held firm with central defenders Reuben Hazell and Aaron Downes on top of their game.
Despite their dominance in the last half hour the best Cheltenham had to show for it in front of goal was one header from Kayode Odejayi which Roche got across to hold comfortably.