Tammy Abraham scored the only goal of the game in the 64th minute as Chelsea won 1-0 at second tier Barnsley to book their quarter-final spot.
Meanwhile, Southampton were also winners on Thursday, scoring two second-half goals from Danny Ings and Stuart Armstrong to win 2-0 at Wolverhampton.
The FA Cup quarter-final draw took place after the matches were concluded with Carlo Ancelotti’s Everton to host in-form Premier League leaders Manchester City next month.
Manchester United will face Leicester City away, while Chelsea hosts Premier League bottom club Sheffield United, with Southampton making the short trip to neighbours Bournemouth.
Chelsea have now conceded one goal in their past five matches in all competitions since Thomas Tuchel took over.
“We clearly can play better, so yes I expected more but not in terms that I am frustrated,” Tuchel said.
“If you lack a rhythm and a bit of self-confidence and you have to show on this pitch, against an opponent that takes all the risk and plays the game of his life, that is not easy. We can do better but I don’t want to be too hard on the players.”
Meanwhile, Southampton boss Ralph Hasenhuttl wasn’t shy about his side’s ambition after reaching the quarters, with a favourable draw against second tier Bournemouth.
“We said when we were more comfortable in the league we can focus on the cup,” Hasenhuttl said.
“We haven’t had an easy time in the last few weeks so it’s important to win and they about what they are doing.
“We want to go for the FA Cup and this is a chance to win something. We have a south coast derby now which is always intense and we will go there trying to win.”