Everton avoided any slip-ups as they beat third-tier Fleetwood Town 1-0 in the Carabao Cup second round to claim their first win of the season on Tuesday.
Frank Lampard's side were one of 10 Premier League clubs in action and edged through to round three thanks to Demarai Gray's 28th-minute goal.
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Fulham were brought back to earth with a bump after their impressive start to their top-flight return.
The Londoners went down 2-0 away to fourth-tier Crawley Town who are 84 places lower on the English ladder.
Manager Marco Silva made 10 changes for the trip and Crawley took advantage when Tom Nichols slotted a 16th-minute opener.
Liverpool loanee James Balagizi made it 2-0 shortly after half-time and the hosts held on comfortably to prompt a pitch invasion after the final whistle.
Leicester City, who like Everton have started the season poorly, gained some relief as they scraped through on penalties against Stockport County after a 0-0 draw.
Premier League Bournemouth required penalties to knock out Norwich City, who were relegated from the top flight last season.
Norwich were poised for the third round as they led 2-1 at Carrow Road, but Brooklyn Genesini equalised in stoppage time before the visitors won the shootout.
Steven Gerrard's Aston Villa won 4-1 at third-tier Bolton Wanderers, Premier League newcomers Nottingham Forest eased to a 3-0 win at Grimsby Town and Wolverhampton Wanderers beat second-tier Preston North End 2-1.
Premier League newcomers Nottingham Forest eased to a 3-0 win at Grimsby Town and Wolverhampton Wanderers beat second-tier Preston North End 2-1.
Crystal Palace scored two late goals to win 2-0 at Oxford United while Brentford were 2-0 winners at fourth-tier Colchester United.
Che Adams maintained his scoring form with two goals as Southampton won 3-0 at Cambridge United.
Adams also scored twice at the weekend as Southampton beat Leicester.
Premier League clubs playing in European competitions are exempt until the third round of the Carabao Cup.