Divock Origi, Roberto Firmino and Sturridge all scored as Liverpool, last season's beaten finalists, crushed Championship side Burton, who are managed by ex-Liverpool midfielder Nigel Clough.
Michy Batshuayi marked his full Chelsea debut with a brace against Bristol Rovers, while Victor Moses was also on target as manager Antonio Conte enjoyed a third straight competitive win.
Premier League sides Crystal Palace, Everton, Hull City, Swansea City and Stoke City also went through, Peter Crouch scoring a hat-trick for Stoke in a 4-0 win at Stevenage.
Liverpool manager Jurgen Klopp handed a debut to Cameroonian centre-back Joel Matip and welcomed back Senegalese winger Sadio Mane following a shoulder injury at Burton's Pirelli Stadium.
Also Read
Mane teed up Origi to break the deadlock with a clever back-heel in the 15th minute before Nathaniel Clyne crossed for Firmino to head in the visitors' second goal.
A Tom Naylor own goal just after the hour made it 3-0 and Sturridge came off the bench to complete the rout with a late brace, tapping in James Milner's cross and slamming home from Mane's pass.
Batshuayi claimed a second from Ruben Loftus-Cheek's cross, but Bristol Rovers scored either side through Peter Hartley and Ellis Harrison to keep Conte's men on their toes until the final whistle.
Disclaimer: No Business Standard Journalist was involved in creation of this content