Blues are due to meet the winners of the second-round tie but their replay on Wednesday night was abandoned due to adverse weather conditions.
They will now try again for a third time on January 4, with Lee Clark's side duly set to meet the victors at St Andrew's on the rearranged date of Tuesday January 7, kick-off 7.45pm.
Crawley and Rovers were goalless at the Checkatrade.com Stadium when the replay was abandoned after 75 minutes due to a waterlogged pitch, with the initial meeting at the Memorial Stadium earlier this month also finishing in stalemate.
It means Birmingham will now enjoy a fixture-free weekend early in the new year, with all other Championship and Premier League clubs in third-round action.
That is sure to provide a welcome boost to boss Clark, whose side already face four games in 12 days starting with Saturday's home encounter against Nottingham Forest.