Clint Dempsey scored twice while Gareth Bale was also on target as Spurs saw off Coventry in the third round on Saturday.
Leeds and Birmingham will meet again after they played out a 1-1 draw at Elland Road.
Spurs needed a replay to defeat the Whites in this competition three years ago while they have only won two of their last six meetings with Blues.
The fourth-round tie will be played on the weekend of 26/27 January.
- Sports & Recreation
- Gareth Bale
- Clint Dempsey