Chris Atkinson and Sam Togwell gave Chesterfield a 2-0 interval lead and although Kaid Mohamed pulled one back, Marc Richards and Jack Lester completed an impressive performance by Paul Cook's team.
Chesterfield started well and Atkinson gave the home side a deserved lead in the 17th minute when Tendayi Darikwa set him up to score with a low right-foot shot from just inside the area.
Darikwa was also involved in the second when his shot came back off the bar and Togwell dived forward to head the rebound in from eight yards.
Cheltenham were back in it after 57 minutes when Mohamed headed in an in-swinging corner but two minutes later Chesterfield scored a third when Atkinson put Richards through to fire over Brown.
The contest was over in the 75th minute when Lester bundled a corner in at the back post to score with his first touch after coming on as a substitute.