Barnet ensured there would be a non-league presence in the fourth round with a 1-0 win away to second-tier Championship club Sheffield United.
Manchester City manager Pep Guardiola may have made eight changes from the team that beat Premier League leaders Liverpool on Thursday but still fielded a strong side for the visit of Championship side Rotherham, with Raheem Sterling, Riyad Mahrez and Gabriel Jesus all starting as playmaker Kevin De Bruyne returned from injury.
Sterling made it 1-0 in the 12th minute and City then scored twice in the closing stages of the first half.
Phil Foden, the England Under-21 international, got on the end of a cross from Ilkay Gundogan in the 43rd minute to score his first home goal for City before Rotherham defender Semi Ajayi turned the ball into his own net.
Jesus made it 4-0 early in the second half before Mahrez, Nicolas Otamendi and Leroy Sane added further goals.
"We are going to try and do well in every competition we play in," Guardiola told the BBC.
Turning to Foden, he added: "He missed some simple passes and he needs to be solid but he is young. When he attacks the central defenders he is incredible."
- 'Phenomenal' Oldham -
Denis Odoi put Fulham ahead in the 52nd minute at Craven Cottage but Sam Sturridge drew Oldham level with a penalty before Callum Lang's header two minutes from time sealed a comeback win.
Daniel Iversen compounded Fulham's misery by saving an Aleksandar Mitrovic penalty just moments earlier.
"It is Roy of the Rovers stuff," said Oldham boss Pete Wild. "The character in the dressing room is phenomenal."
Elsewhere, Barnet beat Championship high-flyers Sheffield United thanks to Shaquile Coulthirst's 21st-minute penalty.
"It's unbelievable -- that's the only word I have at the minute," said Barnet captain Callum Reynolds after the north London club's triumph.
"I mean they (Sheffield United) are third in the Championship and we beat them at their ground."
Blades manager Chris Wilder blasted his side's "arrogance", insisting: "The fans should have booed the players louder."
Watford manager Javi Gracia changed his entire starting side from the team that drew 3-3 at Bournemouth, with Venezuela forward Adalberto Penaranda making his Hornets debut at non-league Woking.
But hopes of Woking, 110 places adrift of the Hornets, enjoying a shock win suffered a setback when Will Hughes curled in a 13th-minute shot from the edge of the area.
Watford made it 2-0 16 minutes from time when Troy Deeney scored from close range.
Millwall came from a goal down to beat Championship rivals Hull 2-1, substitute Shane Ferguson scoring twice inside the closing eight minutes.
QPR defeated Championship leaders Leeds by the same scoreline, Jake Bidwell grabbing a 75th-minute winner for the west London club.
Fourth-tier Newport County are at home to Premier League Leicester in Sunday's 1630 GMT kick-off.
On Saturday, FA Cup holders Chelsea cruised to a 2-0 win over Nottingham Forest in what seems certain to have been Cesc Fabregas's final match for the Blues.
It was the same result as Manchester United beat Reading to make it five wins out of five under caretaker boss Ole Gunnar Solskjaer.
Arsenal, the record 13-times FA Cup winners, won 3-0 at Blackpool, West Ham beat Birmingham 2-0 and Everton defeated Lincoln 2-1.
Saturday also saw Championship club Bristol City beat top-flight basement club Huddersfield 1-0.
Elsewhere, in an all-Premier League clash, Brighton won 3-1 away to south coast rivals Bournemouth.
Second division Derby, managed by ex-England midfielder Frank Lampard, came from 2-0 down to draw 2-2 with Premier League strugglers Southampton.
Newcastle were also held to a draw, 1-1 against second-tier Blackburn.
Premier League leaders Liverpool will complete a third round that started with Tottenham's 7-0 thrashing of Tranmere on Friday when they travel to top-flight rivals Wolves on Monday.