Chasetown actually created a host of early chances but fell behind in the 22nd minute.
A Danny Cocks free kick was cleared by the visitors who broke quickly and Desean Martin fired low past Curtis Pond.
A minute later it was 2-0. Pond made an error outside his area and Martin rolled the ball into an empty net.
The Scholars were denied a penalty at the start of the second half when Jake Lovatt was sent tumbling in the box. Referee Mr Wainman turned away the strong appeals.
Pond made a superb save to keep it 2-0 as Martin bore down on goal for his hat trick.
Drayton keeper Ash Rawlings made a superb fingertip save to deny George Cater from outside the box.
Cocks was then sandwiched between two defenders and knocked to the ground but again the spot kick appeals were waved away.
Substitute Muzzy Nduna almost gave the home side a lifeline late on but Rawlings palmed his low shot wide of the post.