Twelve-time champions United, who beat Huddersfield Town to reach the last eight for a fourth consecutive campaign, are bidding for a second title in the last three seasons.
Brighton, meanwhile, overcame League Two side Coventry City to reach the quarter-finals for the first time in 32 years and are going in search of only a second ever appearance in the final of the competition.
The tie could be one of four all-Premier League contests in the quarter-finals, although the four Champions League-chasing teams to still be involved have all been kept apart.
Manchester City, meanwhile, will continue their pursuit of a quadruple against top-flight opposition in the form of Southampton if they are able to beat Wigan Athletic in their fifth-round clash on Monday night.
Tottenham Hotspur are also yet to play their fifth-round game, but if they can avoid an upset against Rochdale on Sunday then they will face either Sheffield Wednesday or Swansea City, who will contest a replay at the Liberty Stadium following a 0-0 draw this afternoon.
The quarter-final ties will be played from March 16 to March 19.
FA Cup quarter-final draw in full:
Sheffield Wednesday or Swansea vs. Rochdale or Tottenham Hotspur
Manchester United vs. Brighton & Hove Albion
Leicester City vs. Chelsea
Wigan Athletic or Manchester City vs. Southampton