Van Persie scored with United's only two efforts on target to give Louis van Gaal's side its fifth win a row and overtake Southampton in third place. Southampton dropped to fifth behind West Ham after its third straight loss against the teams it hopes to challenge for a top-four spot -- Manchester City, Arsenal and United.
Van Persie punished a wasteful performance from the home side, which created enough chances to win the game comfortably.
"I don't think we played our best football today," Van Persie said. "In the end, it's a fifth straight victory. We're on the way up."
United's opener came when Southampton captain Jose Fonte hit a poor backpass to goalkeeper Fraser Forster and Van Persie ran through, waited for Forster to dive and stroked the ball through his legs and into the far corner.
It finally came after Marouane Fellaini gave the ball away in midfield, and Dusan Tadic initiated a quick attack. He played the ball out to Shane Long on the edge of the area, who returned it with a high cross. Tadic's effort was blocked by Paddy McNair, but the ball fell to Pelle who converted from close range.
Long could have put Ronald Koeman's side ahead in the 34th but blasted high and wide from close range after McNair failed to clear a touch from divert a touch from Sadio Mane. Southampton's dominance was so strong that Van Gaal felt compelled to take off the struggling McNair in the 39th and replace him with Ander Herrera.
