The two-goal cushion lasted barely a minute as Adnan Januzaj curled through a pass that Welbeck seized upon, before chipping the outrushing Hugo Lloris for a fine finish.
Spurs hung on in the final stages, with Lloris' reckless lunge at Ashley Young just inside the penalty area going unpunished. Moyes was incredulous at the decision of referee Howard Webb, saying: "I thought it was scandalous. It's a scandal.''
"You'd hope the people who put the referees out there would have a look at it and look to see if they think the referees are doing as well as they can be,'' Moyes added in a lengthy attack on Webb, arguably English football's top referee.
Spurs had gone 23 years without a league win at Old Trafford until a 3-2 victory last season. Now they have won there two seasons in a row.
Adebayor was carried off on a stretcher and Sherwood said he was unsure whether the striker would be fit to play against former club Arsenal in a north London derby in the third round of the FA Cup on Saturday.
"He has ice on everything,'' Sherwood said, without specifying the injury. "He doesn't know what is hurting him more.''