Joe Quigley gave the Spireites a first-half lead before Rajovic levelled soon after coming off the bench and Dele-Bashiru snatched the late winner.
Watford head coach Valerien Ismael said: "The main thing in the Cup is always to come through, because nobody will talk about the actual game.
"We made the game complicated for ourselves and credit to Chesterfield - they played well with a nothing-to-lose mentality." Defeat was a cruel blow for Paul Cook’s side who deserved at least a replay.
The Championship side gave a first start to Rhys Healey, while Chesterfield made five changes – Liam Mandeville, Ryheem Sheckleford, Joe Quigley, Ollie Banks, and keeper Ryan Boot coming in for the National League leaders.
Watford almost took a fifth-minute lead when Yaser Asprilla intercepted a back-pass from Branden Horton but his effort was blocked by Shackleford.
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