Goals from Morgan Amalfitano and Saido Berahino secured the points for the visitors despite Wayne Rooney netting for the fifth time in as many games. United have now lost three of their first six Barclays Premier League games and have slipped into the bottom half of the table.
"It was a poor result and a poor performance," Moyes said. "We never really got going. We had a lot of the ball in the first half, but didn't make many chances. They always looked a threat on the break. We missed that spark and West Brom deserved [the win], no question. I can't argue with that."
The faith United's supporters vowed to show in their new manager is already starting to fray judging by the boos that accompanied the final whistle.
There could hardly have been a greater contrast between the drab fare served up in the first half and the 10-goal thriller in May that served as a fitting epitaph to Sir Alex Ferguson's reign.
United were on top, as expected, but their efforts were sporadic. Rooney, who retained the captain's armband, had one header saved by Boaz Myhill and sent another soaring over the bar. Shinji Kagawa's scheming presented a chance for Alex Buttner, but his scuffed shot was blocked.
For Steve Clarke's side, Berahino flicked a header just wide from Amalfitano's cross and then fired an effort straight at David de Gea at the end of the first half.
Kagawa was replaced by Adnan Januzaj at the break, but the 18-year-old had no time to make an impact before the visitors seized the initiative.
After stealing possession on halfway, Amalfitano was allowed to stride directly towards goal, brushing off a couple of half-hearted attempts to stop him. Left with only De Gea to beat, he showed admirable composure in lifting his shot over the goalkeeper and into the net.
United's response was swift, Rooney expertly curling home a free-kick from about 25 yards. When Moyes brought on Robin Van Persie a few minutes later, Old Trafford expected a routine end to proceedings.
But, sensing their opponents are not the force of old, West Brom went hunting more. After a couple of close calls, they went ahead again when Amalfitano laid a superb pass into the path of Berahino, who found the bottom corner.
As United tried to get out of jail, Marouane Fellaini had a goal ruled out for offside, Myhill punched clear a late Rooney free-kick and Januzaj flashed an injury-time effort wide.