After a goalless first half, the visitors started the second half on top, finally breaking through in the 57th minute, as the 19-year-old tapped in at the far post for his first goal of the season.
Fulham started the game brightly, and came close to taking the lead in the seventh minute when Bobby Decordova-Reid expertly curled a free-kick over the wall but could not get enough dip on the ball as it struck the crossbar.
The rebound fell to Cyrus Christie who sent a header goalwards, but Alex McCarthy managed to tip the ball over the bar to deny the Fulham defender.
After a poor piece of defending from Fulham as they tried to pass it out from the back, the ball fell to Oriol Romeu in plenty of space in the box, but goalkeeper Marek Rodak was able to dive down and make a good save to prevent Saints taking the lead.
Southampton made the most of their good start to the second half, breaking the deadlock in the 57th minute, with Nathan Redmond breaking down the right to cross for Obafemi who was arriving at the back post and just had to tap the ball into the empty net.
In the 78th minute, Rodak came off his line to make a clever save to deny substitute Danny Ings from close range and keep Fulham in the game.
The visitors had a great chance to wrap up the game, when Ings found himself in a one-on-one with Rodak, but the keeper managed to use his legs to make the save and force the ball behind.