The Pics' were impressive throughout and took a 14th minutes lead when Dave Walker slotted in from Hakeem Islam's pass.
Walker tapped home a second, through Grant Beckett's corner, that doubled the advantage.
Walker's nicked an effort in front of The Scholar's keeper Ryan Price, that bobbled inches wide before Danny Bragoli finish looked to wrap up the win for the visitors, after a superb Josh Craddock flick.
Rushall, who are still in the hunt for league promotion this season, will now be in the hat at FA Headquarters for the third qualifying round, which will be drawn on Monday morning.
Olympic must have been pleased that Chasetown' leading scorer Simon Brown was ruled out with a knee injury and the hosts missed his hold-up play and threat for long periods as Kitching's side produced wave after wave of attack.
The Pics' manager had talked in the build-up to the game about how focused his side were and they looked it, taking the game to their opponents from the first whistle with a pace and panache Chasetown simply could not live with.
Just as it appeared that Rushall had taken the sting out of the tie, Chasetown pulled a goal back. Danny Ashton produced a curling corner that was headed in by the unmarked Gavin Saunders at the far post.