Chelsea proved capable of recovering from going behind to win 2-1 at Watford.
As with Monday's 2-1 defeat of West Ham, a late winning goal from Diego Costa from close range secured all three points for new manager Antonio Conte.
Substitute and summer signing Michy Batshuayi had equalised seven minutes earlier in the 80th minute, also from close range. Etienne Capoue had given Watford a 55th-minute lead.
The result leaves Chelsea in fourth place with six points from two games, behind only City, United and Hull, who won 2-0 at Swansea.
Meanwhile, Manchester City joined Manchester United on maximum points and climbed to first in the Premier League with a convincing 4-1 win at Stoke.
England goalkeeper Joe Hart again remained a substitute but City's season and Pep Guardiola's reign as manager gained a sense of momentum with a third victory in eight days.
Last week's 2-1 defeat of Sunderland was followed by Wednesday's 5-0 Champions League win at Steaua Bucharest. At Stoke, two goals each from Sergio Aguero and Nolito brought City's goals total to 11 in three games.
And Burnley impressively earned their first points of the season by defeating Liverpool2-0.
Their visitors' 4-3 win at Arsenal last weekend suggested they could emerge as title contenders. But for all of their dominance of possession and their many goalscoring chances, they were unable to overcome Burnley's resilience.
Sam Vokes scored from outside the area in the second minute, and Andre Gray added their second in the 37th minute from similar range.
The other matches result:
Tottenham 1-0 Crystal City
Everton 2-1 West Brom
Arsenal 0-0 Leicester