Everton avoided any slip-up as they beat third-tier Fleetwood Town 1-0 in the League Cup second round to claim their first win of the season on Tuesday.
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.
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 also 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 fell behind at third-tier Bolton Wanderers but hit back to win 4-1.
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, their first win in the competition since 2018.
Brentford bounced back from their weekend defeat by Fulham to win 2-0 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 City.
Premier League clubs playing in European competitions are exempt until the third round of the League Cup.