Connor Lemonheigh-Evans gave the Gulls the lead on 11 minutes and Billy Waters added another with 15 minutes remaining.
George Carline and Adi Yussuf both had chances cleared off the line for Town.
The Spireites have now won just one of their last seven and have dropped out of the play-offs with five games to go.
They race Woking away on Saturday next.
Here are Liam Norcliffe’s player ratings from yesterday’s defeat...
The Spireites have now won just one of their last seven and have dropped out of the play-offs with five games to go.
3 / 4