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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.
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.
United are three points behind fourth-placed Chelsea and one ahead of Tottenham Hotspur while struggling Watford stayed 19th on 24 points.
The win restored Liverpool's 11-point advantage over second-placed Leicester City.
The result lifted Watford two places to 18th on eight points from 12 games while promoted Norwich, who last won in September when they upset champions Manchester City 3-2 at Carrow Road, dropped to the foot of the table on seven points.
Man City harvested a near-record amount of goals in destroying Watford to the tune of an 8-0 win at the Etihad Stadium.
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.