The tie between the League Two Sulphurites and the Hatters, currently 12th in the Championship, will take place between January 7 and 10.
Simon Weaver's men progressed beyond the second round for the first time in their 107-year history when they upset League One Portsmouth on Saturday, triumphing 2-1 thanks to Luke Armstrong's first-half strike and Jack Diamond's dramatic stoppage-time winner.
But Monday night's draw at Wembley Stadium saw them miss out on a potentially money-spinning fixture against the likes of Manchester City, Liverpool and Manchester United.
Instead, they will visit 10,365-capacity Kenilworth Road as they bid to make history once again.