Struggling West Ham sensed an upset at Anfield when Jarell Quansah scored an own goal on 21 minutes, but Liverpool were back level at half-time through Diogo Jota.
The Portugal forward scored against shortly after the break before Mohamed Salah put the game to bed with 20 minutes remaining.
Liverpool turned a comfortable win into a hammering with two injury-time goals, Cody Gakpo scoring twice to seal a 5-1 win for Arne Slot’s side.
The Reds will continue the defence of their Carabao Cup trophy against Brighton in the fourth round, while Premier League champions Manchester City visit Tottenham in the standout tie.
Man City edged past Championship side Watford on Tuesday night – just two days after their thrilling Premier League stalemate with Arsenal – while Spurs came from behind to beat Coventry City.
Elsewhere, Arsenal were drawn against Championship outfit Preston and Manchester United will welcome Leicester City to Old Trafford.
Carabao Cup fourth round draw
Brentford v Sheffield Wednesday
Southampton v Stoke
Tottenham v Manchester City
AFC Wimbledon or Newcastle v Chelsea
Manchester United v Leicester
Brighton v Liverpool
Preston v Arsenal
Aston Villa v Crystal Palace
Arsenal proved too strong for League One club Bolton Wanderers at the Emirates Stadium, while United thrashed Barnsley 7-0 to reach the fourth round.
There is an all-Premier League clash between Aston Villa and Crystal Palace while Sheffield United and Stoke City will visit top-division clubs in Brentford and Southampton.
The winner of the postponed fixture between Wimbledon and Newcastle United will face in-form Chelsea, who put five past League Two side Barrow at Stamford Bridge.
Ties for the fourth round of the Carabao Cup will take place the week commencing October 28.
Reacting to the draw, ex-England midfielder Jamie Redknapp said: ‘That’s a great draw!
‘Tottenham and Manchester City had some great games over the years. Spurs have got to go strong, this is a big opportunity for them.’
Slot was impressed with his Liverpool team’s performance against West Ham, who have now lost three of their last four games.
‘If you’re facing a team like West Ham, who have a lot of quality, they can press you. Second half they were also threatening on set-pieces,’ Slot said.
‘Since I’ve been here it’s never been easy, although sometimes the result looks different, but probably that’s the Premier League. Every team has quality players.
‘I do feel like in most parts of the game we had control. We had a large part of ball possession, particularly in the first half.’
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