The £24 million summer signing from Arsenal now boasts four goals in just two starts for his new team as United overcame a resilient display from the newly promoted side to snatch a narrow victory for the second time in as many weeks.
However, Sir Alex was keen to pinpoint the impact that Paul Scholes' second-half cameo had on the rhythm of the game, as he came off the substitutes' bench when his side were still trailing 2-1.
"Absolutely, [Van Persie had a fantastic game]. But when Paul Scholes came on the pitch he changed the game completely," the Red Devils boss told Sky Sports.
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"I thought we were more or less in control in the first half but once we went on the back foot and Van Persie scored a fantastic equalising goal, so you're thinking 'Let's get down to business in the second half and get this game done'.
"But credit to Southampton, they kept putting the ball into the box – and we did prepare crosses towards the back post before game – and they scored two goals from them.
"And then I thought we were well out of it before Scholes came on. His vision and consistent passing gave us complete control in the game and Van Persie has come up with two very, very important goals. He's got four goals in two starts. It's a fantastic statistic and he will only get better.""Also Chicharito came on and made a difference," Sir Alex added of the Mexican, making his first appearance of the campaign. "Stretching [the defence], running in between them and making runs off the ball. I was delighted with his performance."
Despite notching six points from a possible nine, United have been behind in all of their Premier League matches so far and Sir Alex believes that consistency at the back is key to positive results, having been able to call upon first-choice centre-backs Rio Ferdinand and Nemanja Vidic for the first time this season.
He added: "When you play two centre-backs regularly, you get consistent results. Last season our best form came when [Jonny] Evans and Ferdinand played together.
"This season we've had four centre-backs out for the first game of the season. Michael Carrick played centre-back and Antonio Valencia played right-back and that is never going to be the perfect solution.
"Rio was back today. I thought he did better in the second half as he took time to get into the game in the first half. He ended up playing very well for us. Vidic is getting there. If we keep the centre-backs all fit it will give us a good chance."