It will be the biggest match of the Australian-born manager's career, and the biggest club match for his goalkeeper, Socceroo Joe Gauci, who is on loan to Port Vale from Aston Villa.
The English midlands side, who are bottom of third-tier League One, knocked out Premier League Sunderland in the last round to reach the quarter-finals for the first time since 1954.
They appear doomed to be relegated to the fourth tier but Brady, who took over in January, has inspired this extraordinary cup run.
The Newcastle-born 51-year-old left Australia to play soccer in England at 16 and stayed, forging a solid but unremarkable career in non-League and the lower divisions, and doing the same as a coach.
Chelsea, two-time European champions, will inevitably be red-hot favourites to reach the semi-finals but Brady will hope to pull off a stunning boilover.
The other ties are harder to call.
Premier League heavyweights Manchester City and Liverpool have been drawn together with City, last year's beaten finalists, at home.
Premier League leaders Arsenal, who have won the competition a record 14 times and are still in contention for a quadruple, will travel to promotion-chasing Championship side Southampton.
The other tie is an all-Premier League affair. The winners of the remaining fifth round tie between West Ham and Brentford, due to be played Monday night (local time), will host Leeds United.
FA CUP SIXTH ROUND DRAW
Southampton v Arsenal
Chelsea v Port Vale
Manchester City v Liverpool
West Ham or Brentford v Leeds
Ties to be played around the weekend of April 4-5.
