The Cottagers settled the fourth round tie 2-1, with Brazilian Muniz hitting an impressive brace.
Muniz's two goals sandwiched a superb effort from Jonny Smith just after halftime.
However, Fulham were good value for the win, with former Latics loanee Martial Godo also coming close and Issa Diop having a header saved.
Wigan had a second equaliser ruled out for offside, with Ronan Darcy putting the ball inside Fulham's net before the match officials' intervention.
In the end, Fulham were able to hold on to go into the hat for the fifth round draw.
Meanwhile, hopes of the Sessegnon brothers facing eachother were dashed as while Ryan started for Fulham, Latics assistant coach Max Rogers, who was filling in for suspended manager Shaun Maloney, chose to leave Stephen on the bench.