But another newcomer, Crystal Palace, lost 2-0 at Southampton to remain next-from-bottom. Sunderland is lodged to the bottom on one point ahead of Sunday's game against Liverpool.
West Brom fully deserved its first win at United in 35 years, with Saido Berahino clinching victory with a 67th-minute strike from the edge of the box that finished off a flowing team move.
Wayne Rooney's free kick had canceled out Morgan Amalfitano's stylish solo goal, but a United side missing several regular first-team stars with a Champions League match at Shakhtar Donetsk in mind struggled throughout. Moyes' team hasn't scored from opening play since the first day of the season.
City was toasting derby glory last weekend after trouncing United 4-1, but was brought down to earth by the loss at Villa.
City twice threw away the lead given by Yaya Toure and Edin Dzeko before Andreas Weimann struck Villa's winner to leave the visitors without an away win this season.
"We played very well and deserved to win, but in five minutes (when Villa scored twice) we threw away everything good we did in the rest of the match," City manager Manuel Pellegrini said. "The way we lost this game was incredible."
City is still in touching distance of Tottenham, which showed it should be held up as a genuine title contender by holding Chelsea to a draw.
Gylfi Sigurdsson gave Spurs a deserved first-half lead but Chelsea seized the initiative in the second half, with John Terry equalizing with a header from Juan Mata's free kick.
With around 10 minutes to go, Torres, who had been at the heart of Chelsea's second-half attacking threat, was shown a second yellow card for an aerial collision with Jan Vertonghen.
"I'm sad because the game was good," Mourinho said. "In the first half it was not such a good Chelsea, in the second half Chelsea was not good Chelsea was very, very good."
Arsenal can go top by beating Swansea away in Saturday's late game.
Cardiff inflicted more misery on Fulham manager Martin Jol after substitute Jordon Mutch curled home the winner in the second minute of injury time. Steven Caulker's opener for Cardiff in the 12th had been cancelled out in the 45th by substitute Bryan Ruiz.
Brady won and converted a 12th-minute penalty for Hull while Pablo Osvaldo and Rickie Lambert with his 200th league goal were the scorers for Southampton, which climbed to fourth place.
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AP Sports Writer Rob Harris contributed to this report.