Davie struck on 30 and 35 minutes to give the Millers a two goal half time lead despite having been second best to the Scholars for much of the opening period.
Only some fine goalkeeping from Jack Steggles and wayward finishing from the visitors had kept the score goalless before Davie fired past Curtis Pond in the Chasetown goal to open the scoring.
He added a second with a deflected effort after the Scholars failed to clear a throw-in.
Carlton looked much better in the second period and hit the bar twice before Chasetown were offered a lifeline on 67 minutes when Mitchell Piggon lifted the ball over Steggles to reduce the arrears.
The visitors pushed forward for an equaliser but the Carlton defence held firm and Wayne Scott's side secured the win in added time when Jake Wholey fired home after good work by Luke Smithson.
The Millers entertain Southern Premier side St Ives Town in the next round on 29 October.
A report will be on site tonight.