The draw was held on Monday following the conclusion of the fifth round of FA Cup play as Manchester United beat Chelsea to complete the set of eight matches.
Quarter-final matches will be played on the weekend of March 16 with the winners booking a date in the semi-finals at Wembley Stadium on April 6.
Manchester United advanced to the quarter-final after taking down reigning champions Chelsea by a 2-0 scoreline on Monday thanks to goals from Ander Herrera and Paul Pogba.
A trip away to Wolves will be Ole Gunnar Solskjaer's next task after the fellow Premier League side edged Bristol City 1-0 to advance to the quarters.
Manchester United's crosstown rivals, Manchester City, had little problem with League Two side Newport County, as the Premier League title contenders picked up a 4-1 victory away from home.
With that victory, Pep Guardiola's side has now won four straight since a stunning Premier League loss to Newcastle, with the aggregate score of those matches coming to 15-2.
Swansea, along with Millwall, are the lowest-ranking remaining sides in the competition as the six other teams remaining are all representing the Premier League.
Millwall, currently sitting 20th in the Championship, will play host to Brighton & Hove Albion after edging AFC Wimbledon in their fifth-round match. Brighton, meanwhile, held on for a tight 2-1 victory over Derby County in their last match.
Watford currently sit eighth in the Premier League and are fresh off a 1-0 win over Queens Park Rangers while Palace are 13th in the league with a 2-0 win over Doncaster Rovers pushing them to the quarter-final round.
FA Cup fifth round draw in full:
- Swansea vs. Manchester City
- Watford vs. Crystal Palace
- Wolves vs. Manchester United
- Millwall vs. Brighton