Market Drayton were inches away from breaking the deadlock as Lucas Green-Birch struck the outside of Chasetown's right hand post.
Chasetown took the lead in the 37th minute. A cross into the box was fumbled by Matt Ballard and Ryan Wynter fired into an empty net.
The Scholars should have doubled their lead. Build up play between George Cater and Wynter fed Jack Langston but the final pass back to Cater was overhit.
The visitors started the second period on the front foot and, within 30 seconds, a low Cater shot was kicked clear by Ballard.
Wynter almost added his second when Bullard mishit a clearance leaving an open goal. From long range, Wynter’s effort was tipped onto the bar by a retreating Ballard.
On the hour, Langston’s shot was saved by Bullard and then Cater dragged a low right footer just wide of the upright.
Market Drayton pushed for an equaliser without really threatening Curtis Pond into making a save.