Ragnar Klavan was Liverpool's hero with a late winner against Burnley on Monday.
The Reds, without Roberto Firmino and injured duo Philippe Coutinho and Mohamed Salah in their starting line-up, led through Sadio Mane's second-half strike.
Johann Gudmundsson equalised with a late header, but Estonian centre-back Klavan scored in injury time.
Leicester beat Huddersfield 3-0, with Riyad Mahrez scoring and then assisting Islam Slimani, who was making only his second Premier League start of the season, because Jamie Vardy was out injured. Marc Albrighton scored their late third.
Newcastle beat fellow strugglers Stoke City 1-0, with Ayoze Perez scoring from close range.
Brighton and Bournemouth played out an entertaining 2-2 draw in the first Premier League game of 2018 - in a match with 42 shots, the record for a top-flight game this season. The Cherries twice came from behind, with Callum Wilson scoring a scrappy 79th-minute equaliser.
In the 17:30 GMT game, which is on BBC Radio 5 live, Everton are at home to Manchester United.
Premier League match reports
Brighton 2-2 Bournemouth
Burnley 1-2 Liverpool
Leicester 3-0 Huddersfield
Stoke 0-1 Newcastle
Live text: Everton v Manchester United
In the Championship, Daniel Ayala scored twice as Middlesbrough came from behind to beat Preston 3-2 for their first win under Tony Pulis.
Managerless Nottingham Forest, who sacked Mark Warburton on New Year's Eve, drew 0-0 with Leeds.
League Two leaders Luton Town beat promotion rivals Lincoln City in a 4-2 thriller, with both sides having players sent off in the first half.
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