Arsene Wenger's men crashed to their third successive league defeat and fell five points behind fourth-placed Chelsea.
Arsenal trail leaders Manchester City by a massive 18 points but the prospect of failing to qualify for Europe's elite competition next season, a scenario described by Wenger earlier this season as a "disaster," moved a step closer.
Arsenal could secure their place in next season's competition by lifting the European Cup in Munich in May, but that looks highly unlikely after failing to beat the likes of Swansea City and Fulham in the Premier League this month, let alone European giants Barcelona and Real Madrid.
Asked about his team's Champions League prospects, Wenger said: "It leaves us in a very difficult position, it was a match we could not afford to lose today."
Wenger was more angry after he was questioned for his late substitution of 18-year-old Alex Oxlade-Chamberlain, arguably Arsenal's best player who was replaced by Andrei Arshavin, long out of form and favour with the Arsenal fans.
The decision was roundly booed by home supporters and even skipper Robin Van Persie was seen to mouth "no" and shake his head as Arshavin came on.
"I made the decision because he started to fatigue and because Arshavin is captain of the Russian national team and if I have to justify that he comes on a substitute for a guy of 18 years old playing his second or third game, let's be serious," Wenger told reporters.
"People pay for their tickets and are free to express their emotions and we have to live with that, but that doesn't mean they are right.
"I have to stand up for the substitutes I made. I have been 30 years in this job and made 50,000 substitutions and I don't think I have to justify to you every time I make a decision."
Arshavin, though, did turn his back on the advancing Antonio Valencia who provided the pass for Welbeck's winning goal.
"Don't expect me to blame any individual," Wenger said. "I will not do that. We win together and we lose together."
United manager Sir Alex Ferguson was delighted with a win that kept his side within three points of City at the top of the table.
"They had three good chances to score at the start of the second half and missed them all," Ferguson told Sky Sports. "So we rode our luck there, but after that we dominated the game and had some great chances."
He added that it had been vital to match City's win over Tottenham Hotspur.
"That was the most important thing," said Ferguson.
"We played very positive football today, we were adventurous so I am very pleased with that. At this time of the season we have some tough games and now we have four massive away games after this. It was important to win today."
