Arsenal secured a dominant 4-0 victory over Wigan Athletic in the FA Cup, while Leeds United edged Birmingham City on penalties, and Wolverhampton Wanderers battled to a 1-0 win against Grimsby Town.
Brentford scraped into the FA Cup fifth round with a narrow 1-0 win over sixth-tier Macclesfield, sealed by Sam Heathcote's second-half own goal after a spirited display from the hosts.
Manchester City defeated Salford City in the FA Cup, while Burnley suffered a surprising defeat to Mansfield Town. West Ham United and Norwich City also secured victories in their respective matches.
Gabriel Martinelli scored his first hat-trick for Arsenal as the Premier League leaders beat second-tier strugglers Portsmouth 4-1 to reach the FA Cup fourth round.
City, Liverpool all cruise in respective FA Cup matches played on Saturday.
British media said Arsenal had allocated only 8,000 tickets for United supporters, citing safety concerns.
Non-league Macclesfield produced perhaps the FA Cup's greatest upset on Saturday when they beat holders Crystal Palace 2-1 in a thirdround thriller, knocking Oliver Glasner's team out of the competition and sending the Moss Rose crowd into raptures.
Championship side Oxford United beat League Two's MK Dons on penalties after their match ended 1-1, while a couple of League One teams moved on, with Wigan Athletic beating Championship side Preston North End 1-0, and Port Vale defeating League Two's Fleetwood Town 1-0.
'Following successful live operation in the Emirates FA Cup, the Premier League will look to implement the system later this season.'
Images from Saturday's FA Cup quarter-finals.
City then comfortably closed the game out and victory in their record-extending seventh consecutive FA Cup semi-final means they will face Crystal Palace, who beat Aston Villa 3-0 on Saturday, in the final on May 17.
Images from the FA Cup semi-final between Crystal Palace and Aston Villa at Wembley stadium, London, on Saturday.
Everton's Ashley Young came on as a second-half substitute but his 18-year-old son Tyler remained on the Peterborough bench.
Welbeck extra-time winner sends Brighton into quarters
Crystal Palace's Eberechi Eze sparked a massive south London party by scoring the only goal to win the FA Cup 1-0 against Manchester City on Saturday and claim the club's first major trophy in their history.
A much-changed Liverpool were dumped out of the FA Cup by struggling second-tier side Plymouth Argyle, who pulled off a stunning 1-0 fourth round win.
Nottingham Forest reached the FA Cup quarter-finals as they beat fellow Premier League club Ipswich Town on penalties after the City Ground clash ended 1-1 after extra time on Monday.
Some lament the FA Cup having lost much of its romance in recent years, in the face of obsession with the Premier League, but this tie had it in bucket loads.
Reach fifth round for first time in 10 years
A round-up of Saturday's fifth round action in the FA Cup
Manchester City came back from a goal down to beat Bournemouth 2-1 on Sunday to book their spot in the semi-finals of the FA Cup.
Aston Villa striker Marcus Rashford scored twice in a 3-0 defeat of second-tier Preston North End in their FA Cup quarter-final at Deepdale on Sunday to earn a Wembley semi-final against fellow Premier League team Crystal Palace.
Manchester United clawed back from a goal down to beat Leicester City in a 2-1 fourth-round FA Cup victory on Friday at Old Trafford, thanks to a last-gasp controversial header from veteran defender Harry Maguire plus a strike from Joshua Zirkzee.
Joshua Zirkzee scored the winning penalty for 10-man Manchester United against Arsenal after their hectic FA Cup third round tie finished 1-1 after extra time.
Crystal Palace have never won the FA Cup, Aston Villa last lifted the trophy in 1957 and Nottingham Forest have not managed it since 1959 but all three clubs go into this weekend's semi-finals genuinely believing this could be their year.
Manchester United survived one of the most astonishing FA Cup semi-final comebacks ever to beat second-tier Coventry City on penalties after a spellbinding clash ended 3-3 after extra-time at Wembley Stadium on Sunday.
Aston Villa newcomer Marco Asensio struck twice in the second half to lead the Premier League team to a 2-0 victory over second-tier Cardiff City on Friday and into the FA Cup quarter-finals for the first time in 10 years.
A summary of Saturday's action in the FA Cup third round.
Carrick faces first test as United boss in Manchester derby showdown
Former midfielder Fletcher made 342 appearances in all competitions with the club, the majority of them under Ferguson. He was also recently coach of the under-18 squad.
With 17 goal attempts compared to Chelsea's 10, Arsenal will feel they should have gone home with a bigger margin of victory and it will be all to play for in north London on February 3.
Two Bundesliga matches have been scrapped due to large amounts of snow and ice blanketing pitches across Germany, while a third-round FA Cup match on Saturday has also been postponed due to freezing conditions in Britain.
Vitaliy Mykolenko's 39th-minute own goal had put Luton ahead before Jack Harrison cheered the Everton faithful with an equaliser in the 55th minute, hammering in a low shot that Luton keeper Tim Krul allowed to let slip through his hands.
Luton Town knocked out Everton and sixth tier Maidstone United upset Championship Ipswich Town
Holders Manchester City beat Chelsea 1-0 with a late Bernardo Silva goal on Saturday to reach the FA Cup final after an absorbing end-to-end Wembley semi-final left the exhausted winners still chasing a domestic double.
Arsenal manager Mikel Arteta called on his players to "live in the present" ahead of Sunday's visit to local rivals Tottenham Hotspur in a battle to stay ahead in the title race after their lead was cut short by successive Premier League draws.
Arsenal edged past Crystal Palace on penalties on Tuesday to reach the semi-finals of the League Cup.
Championship (second tier) side Rotherham, who haven't made it through to the fourth round of the competition since 2001-02, kept their composure early on as Fulham controlled the play.
Antoine Semenyo has bewildered defences with 10 goals and three assists this season for Bournemouth.
Chelsea had looked in full control at halftime after goals from Marc Cucurella and Cole Palmer