The all-British bout, announced last month, has generated huge interest and has been billed as the biggest in British boxing history.
Around 80,000 tickets went on sale at midday, with prices ranging from £88.50 to £13,128, including booking fees, the BBC reported.
More than 500,000 fans were reportedly in the online queue within minutes of sales opening.
Fury and Joshua are scheduled to headline a December 11th card, although their ring walks are expected to begin at around 1 a.m. on December 12th.
According to an application submitted to Cardiff City Council, Principality Stadium would remain open until 3.30 a.m. on December 12th.
The fight between the former world champions is being broadcast on Netflix, and Saudi money man Turki Alalshikh had hoped initially that it could be staged at New York's Madison Square Garden.
The contest will mark the first meeting between the former world heavyweight champions, ending years of speculation over a matchup that many feared might never happen.