A week on from their 4-1 derby humiliation at Manchester City, United were looking to return to winning ways in the league but instead slumped to a defeat that left the champions in 12th place in the table.
On-loan Marseille winger Morgan Amalfitano put West Brom ahead with an exquisite individual goal in the 54th minute, picking the ball up on halfway and nutmegging Rio Ferdinand before insouciantly dinking the ball over goalkeeper David de Gea.
Wayne Rooney equalised three minutes later, with a free-kick from wide on the left that eluded everyone, only for 20-year-old midfielder Saido Berahino to restore West Brom's lead with a fierce left-foot drive.
Steve Clarke's side held on for a victory that means United have now made their worst ever start to a Premier League season, in Moyes's first campaign since succeeding the legendary Alex Ferguson as manager.
United's fans could nonetheless take a small crumb of consolation from the fact that their conquerors from last weekend fell to a 3-2 defeat at Aston Villa.
Yaya Toure gave City the lead on the stroke of half-time, with Edin Dzeko putting the visitors back in front in the 56th minute after Karim El Ahmadi had equalised for Villa early in the second period.
However, a fine free-kick from Leonardo Bacuna saw Villa equalise in the 73rd minute before Andreas Weimann ran clear and beat Joe Hart two minutes later to give Paul Lambert's side a stunning win.
City had been on course to take over at the top of the table from Tottenham Hotspur, who drew 1-1 at home to Chelsea in the first managerial meeting between former colleagues Jose Mourinho and Andre Villas-Boas.
Torres was shown a second yellow card for leading with his arm in an aerial challenge with Jan Vertonghen, but Mourinho felt the decision was unjust.
The result nudged Spurs to the top of the standings, but Arsenal will replace them if they avoid defeat at Swansea City in the tea-time game.
Record signing Dani Osvaldo scored his first Southampton goal in a 2-0 win at home to Crystal Palace, which sent Mauricio Pochettino's side to the heady heights of fourth place.
An injury-time thunderbolt from Jordan Mutch gave Cardiff City a dramatic 2-1 win at Fulham, while Hull City beat West Ham United 1-0 via a 12th-minute penalty by Robbie Brady. (AFP)
