Manchester City roared back from a two-goal half-time deficit to score three times after the break.
West Brom were relegated from the Premier League on Tuesday after Southampton took a huge step towards keeping their place in the English top flight by beating fellow strugglers Swansea
Premier League: Chelsea, in fifth, are hard on Tottenham Hotspur's heels in the battle for a top four finish.
Manchester City took just 11 minutes to open the scoring as David Silva juggled a Raheem Sterling pass and volleyed home.
Goals from Romelu Lukaku and Nemanja Matic helped Manchester United beat Brighton 2-0.
Swansea's demolition job means boss Carlos Carvalhal can look forward to a fifth round trip to Sheffield Wednesday.
Sergio Aguero's hat-trick inspired a 6-0 demolition of Watford.
Newcastle's second win lifted them into mid-table, providing Rafael Benitez with the hope they can avoid relegation despite their lack of top-class signings since last season's promotion.
Renato Sanches has joined Swansea City on a season-long loan move from Bayern Munich
Attacking changes left Swansea vulnerable at the back in the latter stages and United took full advantage to score three goals in the space of just four minutes.