The win ended the Magic's seven-game home losing streak, their longest skid in Orlando since 2003-04.
Arron Afflalo and Tobias Harris added 17 and 15 points, respectively. The double-double was the 35th of the season for Vucevic.
Thaddeus Young had 26 points and 12 rebounds for the Sixers, followed by Damien Wilkins with 16 points in his first start of the season.
Philadelphia has dropped five straight and 13 in a row on the road.
The Magic next host the Los Angeles Lakers on Tuesday in Dwight Howard's first return to Orlando since his departure this past summer.
The Magic led by just a point entering the fourth quarter and it remained tight throughout.
Orlando got a little separation on a 3-pointer by Nelson that gave the Magic a 91-87 lead with just over 3 minutes to play in the game.
Trailing 91-89, Philadelphia's Jrue Holiday had a chance to tie it at the free throw line, but came up empty on both tries.
Afflalo then knocked down a jumper on the ensuing Magic touch to stretch their lead to 93-89.
Both teams traded misses before Beno Udrih's steal eventually led to a 3 from the corner by Afflalo with less than a minute remaining.
Following a Philadelphia miss, Nelson hit a jumper that made it 98-91 and the Magic closed out the win at the foul line.
After giving Charles Jenkins his first start at shooting guard on Friday since he came over from Golden State at the trade deadline, Sixers coach Doug Collins started Damien Wilkins for the first time. Wilkins made the most of the opportunity, providing an early lift, knocking down open jumpers and getting into the lane.
But Harris' second straight start at power forward also helped give a jolt to the Magic's offense, as he matched baskets with Philadelphia's Young throughout to help spread out the Sixers' defense.
The Magic shot 54 percent (25 for 46) in the first half to take a 56-51 lead into the locker room.
Four players reached double figures for Orlando in the opening 24 minutes, led by Vucevic's 12 points and seven rebounds.
The Sixers also shot above 50 percent (22 for 43) to stay close, including connecting on 4 of 5 from beyond the arc.