The National League leaders held Bristol to a 1-1 draw yesterday, equalising late on after they were forced to play the final 20 minutes a man short due to an injury.
Hayes Lane will host the replay at 7.45pm next Tuesday, with both clubs receiving more than £30,000 for the coverage.
Victory will see Bromley progress to round two for the first time since 1945-46.
They frustrated Rovers for much of yesterday’s game, restricting the League One to just two shots on target.
Adrian Clifton was injured after all three substitutes had been used up, meaning Bromley had to finish the game a man-down, and things became worse when the home-side went ahead on 78 minutes.
Against the odds, Bush hooked home from close range in the 83rd minute to earn the visitors a deserved replay.