Draper intercepted a faulty clearance from Bradford's on-loan defender Rob Kiernan midway through the first half and beat goalkeeper Lenny Pidgeley with a low 25-yard shot into the corner.
Macclesfield then defended doggedly to preserve their lead as Bradford enjoyed 80% of the play but could not find a way through the strong visitors defence.
Midfielder David Syers missed two of their better chances, firing wide in first-half stoppage time after Jamaican international Omar Daley played the ball in to his path and then shot over the crossbar in the 65th minute when debutant right-back Richard Eckersley provided the opening.
Macclesfield goalkeeper Jose Veiga also did well to deny Syers during an intense spell of Bradford pressure late in the game.
Source: PA
Source: PA