Everton will face either Manchester United or West Ham in the FA Cup semi-finals, while Crystal Palace will take on Watford.
The ties will be played on the weekend of 23-24 April at Wembley.
Manchester United will travel to Upton Park for a quarter-final replay after the sides drew 1-1 at Old Trafford.
The date of that last-eight replay will be announced after the second leg of United's Europa League last-16 tie against Liverpool on Thursday.
Everton beat Chelsea 2-0 on Saturday to reach the last four for the first time since 2012.
The other semi-final is a repeat of the 2013 Championship play-off final, which Palace won thanks to a Kevin Phillips penalty.
Watford, who knocked out holders Arsenal on Sunday, will attempt to reach the final for the first time since 1984.
Palace are through to their first FA Cup semi-final in 21 years.