The south Londoners ensured their progression with goals from Marc Guéhi and Michael Olise sealing a victory.
And Palace fans only have a short time to wait before finding out their next opponent, with the fifth round draw taking place on Sunday morning.
FA Cup fifth round draw
Date: Sunday, 6th February
Time: 11:40am GMT
How to follow
The draw will be shown live on ITV from Anfield, before Liverpool host Cardiff City.
The draw will also be covered live across cpfc.co.uk, the official app and the club's social media channels.
What ball number are Palace?
Ball number one.
Who could Palace draw?
Several games are yet to be played, but West Ham United and Chelsea avoided upsets against sixth-tier Kidderminster Harriers and Plymouth Argyle respectively.
Sunday sees Nottingham Forest host Leicester City after their triumph over Arsenal and Bournemouth welcome Boreham Wood.
The following teams have progressed so far:
- West Ham United
- Manchester City
- Norwich City
- Huddersfield Town
- Stoke City
- Crystal Palace
- Luton Town
- Tottenham Hotspur
When is the FA Cup fifth round?
- Fifth round proper: Wednesday, 2nd March 2022
- Quarter-final: Saturday, 19th March 2022
- Semi-final: Saturday, 16th April 2022
- Final: Saturday, 14th May 2022