West Brom lost 4-3 on penalties to Northampton Town following a 2-2 draw and Watford were beaten 2-1 at home after extra time by Gillingham Town, also a third-tier side.
Premier League sides Chelsea, Everton, Stoke City, Crystal Palace, Swansea City and Hull City progressed.
Gillingham came from behind to upset Watford.
Substitute Odion Ighalo gave Watford the lead in the second half but Mark Byrne equalised to take the tie into extra time and striker Bradley Dack grabbed the winner.
Divock Origi and Roberto Firmino gave Liverpool a 2-0 halftime lead at Burton before a Tom Naylor own goal and two late strikes by Daniel Sturridge completed the rout.
"This was a very serious, professional display from our side," Liverpool manager Juergen Klopp said.
"There were nice goals - nothing to criticise. It was humid today, not easy for either team."
Chelsea looked set to cruise into the third round when goals by Michy Batshuayi and Victor Moses gave them a 2-0 lead against Bristol Rovers.
But Peter Hartley pulled one back and although Batshuayi restored the hosts' two-goal advantage before the interval, Ellis Harrison's second-half penalty for third-tier Rovers ensured a tense finale.
Everton forward Arouna Kone scored two late goals to round off a 4-0 win over Yeovil and former England striker Peter Crouch grabbed a hat-trick in Stoke's 4-0 drubbing of hosts Stevenage.
The top-flight teams qualified for European competition will enter the League Cup in the next round.
