Naby Keita scored his first Liverpool goal but it was Mohamed Salah and Jordan Henderson who boosted the Reds' Premier League title hopes.
A 1-0 win at Brighton prompted an emotional reaction from Ralph Hasenhuttl, who apologised for not immediately shaking Chris Hughton's hand.
Pierre-Emile Hojbjerg's second-half effort was the difference as Southampton eased their relegation fears against Brighton and Hove Albion.
Yan Valery has scored in consecutive Premier League appearances and Southampton have handed the 20-year-old defender a new contract.
A late Romelu Lukaku goal denied Southampton a point at Old Trafford, but boss Ralph Hasenhuttl had only praise for his side.
Fulham slid deeper into relegation danger with defeat at Southampton and Claudio Ranieri was left to consider his future at the club.
Arsenal went fourth in the Premier League by beating Southampton, but Unai Emery was keen to temper expectations over his side.
Burnley were denied what looked a blatant penalty – but did score from the spot late on – in their 1-1 draw against Southampton.
Wilfried Zaha scored and was sent off as Southampton came from behind to draw with Crystal Palace.
Southampton boss Ralph Hasenhuttl felt his team could have scored "four or five" during their valuable home victory over Everton.