New manager Bob Peeters had named a mix of senior and development players for the match against the Addicks' neighbours, with new signing George Tucudean making his first appearance on English soil.
Charlton had the better chances throughout, with striker Simon Church narrowly missing on a couple of occasions.
United's defence was solid, though, and they got the match-winner in the second half after they were awarded a penalty for a foul on skipper Daryl McMahon.
Fleet striker Matt Godden made no mistake with the resulting spot-kick, which proved to be enough.
Peeters confirmed afterwards that he would be speaking with Iran international Reza Ghoochannejhad about his future on Thursday following the striker's return from World Cup duty.