Victory would have taken Man Utd into third, but Obafemi bundled the ball home at the far post after Jan Bednarek had flicked on a corner.
Stuart Armstrong gave Saints an early lead after good work by Danny Ings and Nathan Redmond.
But Anthony Martial set up Marcus Rashford to equalise, before driving a fierce shot into the roof of the net three minutes later to put United ahead.
Rashford was denied a second goal midway through the second half by Ryan Bertrand's intervention, but the visitors rallied.
David De Gea's fingertip save kept out Redmond's curled effort late on but, after United substitute Brandon Williams had to go off with a head injury, Obafemi pounced from close range.
The draw keeps United in fifth, level on 59 points with Leicester City.
Southampton remain in 12th on 45 points.
Man Utd: 16 Jul v Crystal Palace (A)
Southampton: 16 Jul v Brighton (H)