The Reds failed to beat the Gas in their two League Two meetings last season.
Joey Barton’s charges, who secured promotion on the last day of the campaign, recorded a 1-0 home over Crawley last September before beating the Reds 2-1 at The People’s Pension Stadium in March.
First round fixtures are due to take place on the week commencing August 8.
The draw was conducted live on Sky Sports News by former Nottingham Forest and Tottenham Hotspur defender Michael Dawson and ex-Crystal Palace and Republic of Ireland striker Clinton Morrison.
The fixture will become Crawley’s third of the campaign, with a League Two trip to Carlisle and a home game against Leyton Orient coming beforehand.
The Reds exited the competition in the first round last season, having lost 11-10 to Steve Evans’ Gillingham.
Ticket information and an exact date for this fixture will be confirmed in due course.