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A second-half goal from Emile Smith-Rowe gave them a 1-0 home win over Watford.
Chelsea manager Thomas Tuchel had called on his attacking players to step up after injuries to striker Romelu Lukaku and Timo Werner.
The 41-year-old Spaniard became the first Premier League manager to lose his job this season.
Tottenham Hotspur maintained their 100 per cent start to the Premier League season with a 1-0 win over Watford.
United are three points behind fourth-placed Chelsea and one ahead of Tottenham Hotspur while struggling Watford stayed 19th on 24 points.
On the evening when the European champions were expected to complete the best sequence in the history of English football's top-flight with a 19th straight league victory, they were instead hammered 3-0 by the team lying one from bottom.
It was a good result for Wolverhampton Wanderers too as they will now qualify for the Europa league via their seventh-place finish in the Premier League.
Arsenal are ahead of fifth-placed Chelsea, who have played a game more, on goal difference.
The win restored Liverpool's 11-point advantage over second-placed Leicester City.
Arsenal squandered a two-goal lead as Watford hit back to draw 2-2 in new manager Quique Sanchez Flores's first game of his second stint in charge at Vicarage Road on Sunday.
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