Frank Lampard's struggling side were one of 10 Premier League clubs in action across 21 ties and edged through to round three thanks to Demarai Gray's 28th-minute goal.
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 halftime 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.