Nigeria’s Tammy Abraham shot Chelsea into the quarters with a lone goal victory over hardfighting Barnsley scored in the 64th minute through Reece James assist.
Earlier Southampton buffeted Wolves 2-0 to also qualify.
Danny Ings supplied the first goal in the 49th minute while Stuart Armstrong scored the second through Nathan Tella assist at 90th minute.
Everton will welcome Premier League leaders Manchester City to Goodison Park in the FA Cup quarter-finals, while Manchester United travel to Leicester.
Chelsea's reward, should they get past Barnsley on Thursday evening, is a home tie against the Premier League's bottom side Sheffield United.
Championship side Bournemouth, who are currently looking for a new permanent manager, host south coast neighbors Southampton.
Everton reached the last eight after coming through a nine-goal thriller against Tottenham on Wednesday, with Bernard's extra-time winner giving them a 5-4 victory over Jose Mourinho's side.
City, meanwhile, made English football history by breaking the record for consecutive wins (15) in all competitions by a top-flight team as they beat Swansea 3-1 in the fifth round on Wednesday.
Everton are yet to face City this season after the original league fixture on December 23 was postponed following a coronavirus outbreak in the City squad. The rearranged match will take place on February 17.
Leicester and United met in the Premier League on Boxing Day last year, when Axel Tuanzebe's 85th-minute own goal earned a point for the Foxes in a 2-2 draw at the King Power.
Just two points currently separate the two sides in the league table after 23 games, with United ahead in second and Leicester third.
The quarter-finals are due to take place on the weekend of March 20-21.
FA Cup quarter-final draw:
Everton vs Manchester City
Bournemouth vs Southampton
Leicester vs Manchester United
Chelsea vs Sheffield United