Tottenham also enjoy a commanding win
Four top-flight sides have avoided EFL Cup exits at the hands of lower league opposition.
West Ham scored the only goal in the 96th minute of their win over Accrington Stanley at the London Stadium through talisman Dimitry Payet, while Manchester United finished 3-1 up at Northampton.
The scores in that latter game had been locked at 1-1 after a penalty from Alex Revell cancelled out Michael Carrick's opener for Man United.
But an Ander Herrera strike from outside the box followed by an error from Northampton's keeper Adam Smith allowed Marcus Rashford to make it 3-1.
Sunderland won 2-1 at QPR thanks to a double from ex-Man United defender Paddy McNair, while Tottenham got past Gillingham by a 5-0 margin with Christian Eriksen scoring twice. Summer signing Vincent Janssen also netted his first Spurs goal from the penalty spot.
Elsewhere, holders Manchester City won 2-1 at Swansea with Gael Clichy and Aleix Garcia's goals proving enough.
Southampton saw off Crystal Palace 2-0, and Hull got a last-gasp winner in their 2-1 victory at Stoke.