Southampton lost 2-0 at home to Wolverhampton Wanderers with Danny Batth and Donovan Wilson on target for Wolves, who won the League Cup in 1974 and 1980.
Newcastle, after two straight losses in the Premier League, were pushed into extra time by Nottingham Forest with the tie ending 2-2 after normal time, Jason Cummings scoring twice for the visitors.
Seven minutes into extra time Tyler Wilson headed in the winner for Forest, four-time winners of the competition, to add to the woes of Newcastle boss Rafa Benitez.
First-half goals from Jack Cork and Robbie Brady gave Burnley a two-goal advantage at the break and the Clarets comfortably saw out the game.
The match between the two old rivals was marred by a Blackburn fan invading the pitch and physically confronting the Burnley players, grabbing the neck of midfielder Ashley Westwood before he was removed by stewards.
Premier League West Ham enjoyed a 2-0 win at League Two Cheltenham Town while Joe Allen scored twice for Stoke City in their 4-0 win over League One Rochdale.
Huddersfield Town have enjoyed life in the Premier League so far with two straight wins and they continued their positive form, coming back from a goal down to win 2-1 at home to Rotherham.
(Editing by Clare Fallon)
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