The 20-year-old's late strike handed the defending champions a crucial three points as they stayed in touch at the top with a performance that "delighted" their Scottish boss
Welbeck struck the decisive blow after Robin van Persie had cancelled out Antonio Valencia's first-half header and, according to his manager, he could have had more than one goal to his name.
"He's fantastic," Sir Alex said of Welbeck to Sky Sports.
"His work rate, his movement, all the time he's on the shoulder of the defender and ready to run through and really he could have scored five today.
"I think he was through on goal four times. So he’s been unlucky today, but he's got an important goal, he's continued his run of goal scoring in the first team this season – I think that's nine goals now – for a 20-year-old, he's doing really well."
Sir Alex expressed his surprise at the number of chances his side were able to create at the Emirates, but admitted they also had to withstand some pressure in their own half at times.
"We've had an amazing number of chances," he said.
"The Emirates is always a difficult place to come to – Arsenal are a top side, and to create that number of chances and only score two was a bit disappointing.
"The first half, in particular, was a golden period for us. Then, in the first 15 minutes of the second half, they've had three great chances and missed them all, so we rode our luck there. But on the counter-attack, in particular, I thought we had some great chances.
"They put [Per] Mertesacker up there near the end and kept tossing the ball into the box at us, and you never know what’s going to happen. He’s 6’ 4”, he’s going to have an advantage over most of us.
"But we coped with that, Chris Smalling was fantastic at that point, and we managed to see it out."
Arsenal manager Arsene Wenger was forced to field centre-backs Johan Djourou and Thomas Vermaelen at full-back against United due to his long injury list but Sir Alex was keen to point out that the Gunners boasted the joint-best home defensive record in the league prior to this fixture.
"That's been a problem for them for a few weeks," he said.
"With [Bacary] Sagna, [Kieran] Gibbs and their natural full-backs out they're having to play centre-backs there, so that's difficult for them.
"But you look at their home record, and their goals against, and they've only lost six goals at home up until today. So they've not done a lot wrong defensively; we've had to play well today, and for most of the game we did play well."
The victory was made all the more important, Sir Alex agreed, following Manchester City's last-gasp win over Tottenham earlier on Sunday.
"We said that before the game – matching Man City was the important thing," he said.
"But I think we've done it the right way – today we've played really adventurously, we were positive and we had good belief in ourselves.
"I'm delighted with that. At this time of the season they're all important games, and I said before the Arsenal game that we've got four massive away games. That's one out of the way."