After a frustrating first half in which the Red Devils found it hard to break down the visitors' solid defence, the hosts then fell behind early in the second half when Jamie Mackie beat Anders Lindergaard.
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Anderson came off the bench to give the home side a bit more energy in midfield, with the Brazilian also assisting the third goal with a superb pass to Chicharito.
Sir Alex applauded the Brazilian for injecting life into the rest of the players, while he also paid tribute to their opponents for showing determination that comes from losing a manager.
"Anderson made a difference, there's no doubt about that," he told Sky Sports. "We expected that QPR would have a sudden surge of determination and enthusiasm. It happens all the time when a team loses its manager. That's exactly what we saw today.
"There were very determined, they worked hard. They played good football at times and had some good counter-attacking play but I thought we were a little lethargic.
"It was too easy to give away possession in the first half in particular but when they scored, Anderson came on and we came to life."
In other good news for the 70-year-old, Darren Fletcher made his first Premier League start since November 2011, and capped the occasion with the Red Devils' crucial second goal.
The long-serving manager was elated for the Scottish midfielder, while he also praised Jonny Evans, who netted the equaliser, for his growing goal threat.
"It's been a long road back for the boy and it's been the first time we've played him twice in a week," he added.
"Normally we give him that seven day break. [He] never lets us down, he always puts in a real trojan's work.
"His work rate is fantastic, he's back to that now. I think we're all pleased for him getting a goal.
"Jonny's got three goals for us from that situation and he's capable of that but it's just now this season that he's emerged as a real threat in the box."
The west London outfit were under command of interim boss Eddie Niedzwiecki, following Mark Hughes' sacking on Friday, but were under the watchful eye of Harry Redknapp, who was announced as the permanent replacement earlier on Saturday.
Sir Alex is confident that the former Tottenham boss can have an impact at the club, but insisted his own side were worth their win this time.
"I think so," he explained. "It's obvious that happens. It's a strange thing, you change your manager and you get a couple of results. They were very close today but once we started to play there was no question over who was going to win."