Drury's fourth-minute free-kick was tipped over by Addicks goalkeeper Rob Elliot but it was Charlton who broke the deadlock two minutes later when Joe Anyinsah headed home Kyel Reid's cross.
Matthew Barnes-Homer's long range drive hit the post but Luton's pressure was rewarded on the half-hour when Simon Francis tripped Amari Morgan-Smith in the area.
Drury saw his penalty saved by Elliot but he was ideally placed to head in the loose ball past the stranded keeper.
Charlton were back in front four minutes later after Mark Tyler could only push out Scott Wagstaff's shot as far as Johnnie Jackson, who headed past the Luton stopper.
Morgan-Smith squandered a glorious chance for the visitors when he headed wide from Matthew Barnes-Homer's 69th minute cross. But the Blue Square Bet Premier side earned a replay seven minutes from time as Drury grabbed his second of the game with a curling 25-yard shot.