The Reds fell behind after six minutes of play when Ashley Westwood’s corner from the left flew straight in, though Alisson Becker felt he was unfairly impeded by James Tarkowski.
Roberto Firmino equalised on 19 minutes, however, with Tom Heaton and Tarkowski failing to deal with a Mohamed Salah cross, allowing the Brazilian to tap in from close range.
Soon after, Sadio Mane fired the hosts into the lead when he rasped a loose ball into the corner with aplomb after Salah had been challenged by Charlie Taylor.
Firmino slotted in a third in the 67th minute, but Burnley reduced the arrears in stoppage-time through substitute Johann Berg Gudmundsson.