George Burley's team won with goals by Adam Bolder, Ian Taylor, Morten Bisgaard, Tommy Smith and Junior. Former Derby player Ian Evatt scored Chesterfield's consolation goal.
Derby's Midlands rivals Nottingham Forest were held goalless at Lincoln, while fellow Championship side Watford surprisingly lost 2-1 to Colchester. Craig Fagan scored both goals for the Us at Layer Road while Danny Webber replied for Watford.
Rotherham won 2-1 away at League Two side Boston but West Ham were held 1-1 by Peterborough at London Road.
Leicester's new signing David Connolly formed an excellent partnership with Dion Dublin in a 2-0 win over St Johnstone. Dublin scored City's second goal from a Connolly pass in the 62nd minute. Leicester's first goal came midway through the first half as Lilian Nalis and Jason Wilcox combined to set up James Scowcroft.
Wolves were held to a goalless draw at Northampton.
Blackpool lost 2-0 at Darlington, our next opposition on Tuesday.