However, they were outplayed for much of Monday's game and only had three shots to the Saints' 15.
United's boss also praised the contribution of Van Persie - who scored either side of Graziano Pelle's equaliser.
Van Gaal had been saying he had a "very bad game" but called his performance at St Mary's his best of the season.
"I think not only the goals are telling something about Robin van Persie, I think also his performance," he said.
"He was one of three players on the pitch for Manchester United who were good or maybe very good. He had a great influence on the result and until now it is his best performance."
After an indifferent start to their season United now find themselves in third position but Van Gaal admitted it was an unsavoury way to achieve it.
"It is fantastic to be third in the table but I had hoped to do it with a better performance," he said.
Shot-shy Manchester United
Manchester United had just three shots at goal at St Mary's, their lowest tally in a Premier League game since 2003.
Southampton, who slip to fifth, have now suffered three successive defeats and manager Ronald Koeman said his side were guilty of not being clinical enough.
He also admitted Van Persie was ultimately the difference between the two sides, adding "the better team" lost.
"We did not lose to Man Utd, we beat ourselves," Koeman told BBC Sport. "We were the better team.
"If you give players like Robin van Persie those chances, he will punish you. We did not punish them and that was the difference.
"We did everything, we created chances, but we made mistakes as well.
"Maybe they made more mistakes than us, but they have players who handle those kinds of mistakes, and that was the difference. Because on football and chances, I think we deserved more."
source: BBC Sports
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