First-half goals from Alex Whittle and Kabongo Tshimanga secured the victory after the hosts had Kyle Storer sent off.
Town had chances to add to their lead and restricted the Moors after the break to bring the three points back to Derbyshire.
The win means the Blues have lost just one in 16 games and are unbeaten in nine.
Here are Liam Norcliffe’s player ratings from the match...
The win means the Blues have lost just one in 16 games and are unbeaten in nine.