Spanish teenager Brahim Diaz scored in each half with his first senior goals to set up a last-eight meeting with Leicester or Southampton.
But there was concern before the end for De Bruyne, who grimaced as he limped off three minutes from time.
The 27-year-old Belgian midfielder was making only his second start of the season after suffering a ligament injury to his right knee in August.
“He is being checked by the doctors,” said City manager Pep Guardiola. “We don’t know if it’s nothing or if it’s something serious.”
De Bruyne had put in a typically impressive and energetic performance but went off after going to ground clutching his left knee, and disappeared straight down the tunnel for treatment.
The Premier League leaders had made 10 changes from the team which beat Tottenham on Monday but still outclassed a full-strength Fulham side, managing 27 shots to five from the visitors.
Diaz broke the deadlock after 18 minutes, meeting Vincent Kompany’s knock-down with a first-time shot that took a huge deflection and span past Sergio Rico.
He added his second on 65 minutes after Rico had tipped a Gabriel Jesus shot on to the post, with the 19-year-old following up to hammer his shot into the roof of the net.
City will find out their quarter-final opponents on 27 November, when Leicester host Southampton.
Their fourth-round tie was originally meant to be played on Tuesday but was postponed after Saturday’s helicopter crash at King Power Stadium that killed Foxes owner Vichai Srivaddhanaprabha and four others.
De Bruyne’s industry and effort had stood out on an otherwise low-key evening at Etihad Stadium, as he roamed across the pitch and helped set the tempo as City took charge.
He had played only 150 minutes in total in 2018-19 before this game but did not look tired, as he continuously drove his side forward and also worked hard to win the ball back.
With seven tackles and three interceptions, De Bruyne played a big part in pegging Fulham back in their own half, and the Cottagers – 18th in the Premier League – often struggled to clear their lines.
His forward play, particularly his crosses from the right, also caused the visitors problems as City created enough chances to win by a much bigger margin.
“World class is something that is bandied about easily but De Bruyne was phenomenal.
Fulham avoid heavy defeat but still make limp exit
The Cottagers arrived at Etihad Stadium with the worst defensive record in the top flight, and with the future of manager Slavisa Jokanovic in doubt.
Given City’s obvious attacking power, the fear for Fulham fans was that the Premier League leaders would overwhelm them, just as they did on their last visit here in September.
City took the lead after only 96 seconds on that occasion but, although they had to wait a little longer to make the breakthrough this time, Fulham again ended up comfortably beaten.
The Cottagers’ resolve to play out from the back was admirable but it frequently put them in trouble and their reluctance to fire the ball up to lone striker Aleksandar Mitrovic left him feeding on scraps.
If it was understandable that Fulham were outclassed by City in every area, there was still a feeling they could have put up more of a fight.
They will have to offer far more resistance, not to mention attacking threat, when they return to Premier League action and visit fellow strugglers Huddersfield on Monday in what already appears to be a key relegation clash.
Man of the match – Brahim Diaz (Man City)
De Bruyne was a delight to watch but Diaz had a night to remember on only his second start of the season. His future at the club is still in doubt but it was clear how much he and his team-mates enjoyed his goals.
What they said
Manchester City boss Pep Guardiola: “We made an excellent performance without the ball in terms of how aggressive we were. They have the quality to play even through us, even though they are not in a good moment in terms of results. We played like the game deserved. We made the space and created enough chances. I’m so satisfied.”
On difficult selection dilemmas: “There are no words to explain how tough it is. I explained it to them. They make my job so easy on one side but at the same time it’s so difficult. Fabian Delph doesn’t play, Danilo is so clever. Gabriel Jesus fought for every ball. I said some names but I could say 11.
“There are a lot of competitions and winter is coming, which is so complicated. Everybody is going to play. They can be upset with me, no problem when they react like that on the field.”
Fulham manager Slavisa Jokanovic: “We wanted to respect the competition and try to go through. We made some positive things. Our defensive organisation was better than the past. They showed their quality and their power and their speed.
“I believe in myself and I must believe in my team. It is not an easy job ahead of us. We are fighting not to get relegated. We are in a situation. Our supporters expect better and we are trying to find solutions.
“This game showed we can make fewer mistakes and be better without the ball. I have seen some positive light ahead of us.”
City continue dominance over Fulham – key stats
- Manchester City have reached the League Cup quarter-finals in consecutive seasons for the first time since 1977-78/1978-79.
- Fulham have lost five consecutive matches in all competitions for the first time since December 2013 (a run of seven).
- City have won their past seven meetings with Fulham in all competitions, by an aggregate score of 21-3.
- In all competitions, City have scored in each of their past 23 games against Fulham – only against Portsmouth (25 between 1963 and 2006) have they had a longer such run.
- City striker Gabriel Jesus attempted eight shots in this match, twice as many as any other player. However, just one of these efforts was on target.
City’s run of five games in 13 days continues with a home Premier League match against Southampton on Sunday (15:00 GMT), and they are in action at Etihad Stadium again on Wednesday when they host Shakhtar Donetsk in the Champions League (20:00 GMT)
Fulham will resume their quest for a first league win since 26 August when they travel to bottom side Huddersfield on Monday (20:00 GMT)