Spireites fans have now snapped up all the tickets for the third round clash on January 8.
The remaining tickets went on general sale at midnight on Wednesday and were sold-out within minutes.
Season ticket holders have been busy buying theirs in the last week.
Anyone who has experienced an issue with their online purchase should email [email protected]
"To ease the burden on ticket office staff, please do not email with any other queries about Chelsea tickets at this time,” the club said.
The much-anticipated tie has been the talk of the town since the draw was made.
It will be the first time the two teams have met since 1950.
Chelsea, who won the Champions League last season, are currently third in the Premier League, six points behind leaders Manchester City.