With one match remaining, Sunderland are four points clear of third-from-bottom Wigan, whose game in hand is against Arsenal on Tuesday. Should Wigan lose, they will go down with Sunderland staying up.
Phil Bardsley looked like giving Paolo Di Canio's side victory - and virtually guaranteeing their safety - when he scored a deflected shot from the edge of the area in the 68th minute.
However, Puncheon hooked the ball into the net from close range after having an initial attempt saved only two minutes after coming on as a substitute to claim a draw for Southampton, who are five points ahead of Wigan.