Marcelo Bielsa's men became only the second top-flight team to lose by at least three goals to a fourth-tier side in the competition since the English Football League added the bottom division in 1958.
The last side to suffer such a result was Oxford United, who lost by the same scoreline against Aldershot Town in January 1987.
Crawley, sixth in League Two, had lost its three previous FA Cup matches with Premier League teams, against Manchester United, Stoke City and Reading, while it had not reached this stage of the competition in eight years.
Nick Tsaroulla scored a fine opener 50 minutes into what was just his seventh appearance in senior football, and Ashley Nadesan doubled the lead three minutes later following an error by goalkeeper Kiko Casilla.
Jordan Tunnicliffe netted a deserved third for Crawley, for whom Wright, best known for his work in English reality TV series The Only Way Is Essex, made a late substitute appearance.
Leeds has now been knocked out of the cup at this stage for the past four seasons, its worst run since another four-campaign stretch from 2003-2004 until 2006-2007.