Scott Parker's side struggled to deal with Southampton's pressing in the first half, regularly getting caught in possession and muscled off the ball, and their lack of firepower was once again apparent as they went in a goal down at the break.
Their lack of creativity was punished on the hour mark when Tella rifled a pass from Kyle Walker-Peters into the net to notch his first senior goal.
Making his first start for Fulham, teenaged winger Fabio Carvalho pulled a goal back in the 75th minute when he thumped home Bobby Reid's pass. But Theo Walcott killed off any hope of a comeback with a deflected shot from Tella's pass in the 82nd minute that flew into the net.
The win leaves Southampton 13th in the table on 43 points, two behind Wolverhampton Wanderers, while Fulham remain 18th on 27 points.