Manchester City’s bid to retain the Carabao Cup will take in a trip to Leicester or Southampton after this canter of a win over Fulham that featured Brahim Díaz’s first goals for the club.

Pep Guardiola will be pleased with securing a place in the quarter-finals but concerned to see Kevin De Bruyne limping off towards the close with what looked a problem with his left knee.

For Fulham the search goes on for a victory – or a draw – this being a fifth consecutive defeat for the Premier League’s third-bottom team.

Guardiola retained only John Stones from Monday’s 1-0 win at Tottenham. Phil Foden made a second start in the League Cup and De Bruyne only a second following a three-month absence because of a knee injury.

Slavisa Jokanovic made three changes – Tom Cairney, Luciano Vietto and André-Frank Zambo Anguissa came in – with the manager desperate for a win after losing the last four.

De Bruyne took only moments to show his class with a shot and then a cross-shot that Sergio Rico was grateful to gather.

Leroy Sané capitalised on a loose ball to race down the left but when Gabriel Jesus was picked out he failed to control the ball. Sané combined with De Bruyne as the former tore behind the defence and fed the midfielder whose backheel went close.

Fulham were very open and sloppiness already appeared to be City’s enemy. It was too easy. This time Sané fed Foden and his fierce shot was beaten away. De Bruyne’s follow-up was deflected for a corner.

City’s lead came when the ball broke from a corner it bounced off Vincent Kompany’s left knee and Díaz’s hunger allowed him to beat Timothy Fosu-Mensah to the ball and hit home.

Danilo skated along the right and was allowed to career into a shooting position all too easily and only Rico’s left post kept out the right-back’s blazing shot.

City had Fulham were they love to have all opponents: pinned in and around the area facing constant blue waves of attack. De Bruyne, Danilo, Díaz, and Fabian Delph were a dizzying blur of interchanging passing .

When Foden laid off a simple ball to Sané rather than shoot it was an impressive example of sound decision-making by the 18-year-old midfielder.

City, though, did ease off. Passes went awry and their pace sagged. This had Guardiola hanging his head and, for all their dominance, City were only a goal up at the break.

They started the second half with intent. Jesus’s foray took him into Fulham’s area and when the move reversed Delph was fouled in free-kick range. Danilo stepped up but did not trouble Rico.

Aleksandar Mitrovic gave City a scare when he breached their penalty box – a virtual collector’s item – before Foden broke down the left. After swapping passes with Oleksandr Zinchenko he put in Jesus but a player with only one goal in his past 11 appearances showed why, proving unable to at least pose Rico a question.

The match was played at an under-capacity Etihad Stadium with a flat atmosphere. City had caught the mood and a passage of prolonged Fulham pressure showed they retained hope.

Díaz still sparkled, though. One moment he was back helping to defend, the next performing a soft-shoe shuffle to create space for a shot. It was the Spaniard who doubled City’s lead. An injection of what had been missing – speed – from Sané led to Foden touching on to Jesus and after his attempt came off Rico’s left post Díaz smashed home.

Sané was back to the Sané of earlier: driving at Fulham and causing them headaches. Foden again showed smartness with a dinked pass that put Sané in and it began to feel a long way back for Fulham.

With 13 minutes left Riyad Mahrez replaced the excellent Díaz. The forward received an ovation plus a hug from Guardiola, who brought on the 18-year-old French midfielder Claudio Gomes for his debut.

MANCHESTER CITY (4-3-3): Muric, Danilo, Stones, Kompany, Zinchenko, De Bruyne (Gomes 86), Delph, Foden, Diaz (Mahrez 78), G Jesus, Sane. Subs (not used): Ederson, Walker, Bernardo Silva, Sterling, Aguero.

BOOKED: Kompany.

FULHAM (4-2-3-1): Rico; Fosu-Mensah, Odoi, Ream (Le Marchand 69), R. Sessegnon; Anguissa, Seri, Vietto (Christie 90), Schurrle (Ayite 81), Cairney; Mitrovic. Subs (not used): Fabri, Mawson, S. Sessegnon, de la Torre.

REFEREE: Martin Atkinson (West Yorkshire).