The striker hit a superb hat trick to give Manchester United a hard-fought victory at St Mary's Stadium, but he was not satisfied after his missed penalty.
The Dutchman's superb treble was the difference between the two sides, but he was left fuming after attempting to chip a second-half spot-kick that Kelvin Davis palmed away.
He told Sky Sports: "It means a head-ache, I don't know what I was doing with the penalty.
"I wanted to hit it hard, like I always do, at the last second I just changed, somehow. Maybe it was my brother or something, I don't know! But it wasn't good enough, so it was a bit of a downer. Afterwards, we got a bit lucky for 2-2 and at the end it was dramatic for 3-2.
"I'm quite disappointed about this, to be fair. I ask a certain standard of my game and when that happens, you're 2-1 down, you can't take a penalty like that so it's me to blame. I don't know what went wrong, something went wrong big time so I need to work hard."
United was shook by another superb Saints display against a title contender, after its similar opening-day defeat to Manchester City, and while Van Persie hit the goals to seal victory, he was insistent that his side's comeback from 2-1 down was a team effort.
The 29-year-old also reserved special praise for Paul Scholes, who was introduced to the game as a second-half substitute.
He added: "Everyone has to work hard. I need to say a big thank you to Paul Scholes, because when he came on everything was ticking and every single pass he hit was the right one.
"Everyone felt that, I certainly did. With him you're always on your toes because anything can happen with him. For me, he's the man of the match today."
The Dutchman once again hailed everyone involved with the club for making him feel welcome, after his move from Arsenal this summer.
"I have to thank everyone, it's a great club," Van Persie said. "Everyone's been very helpful. I've been here only for two weeks now but I feel very welcome. Everyone that's involved, the players and the staff makes me feel welcome, everyone wants to help and I appreciate that they are trying to help."
Follow GOAL.COM USA on