After a very poor performance for most of the first half from United, they managed to get the crucial breakthrough goal just before half-time when Marouane Fellaini headed home a cross from Bastian Schweinsteiger.
In all it meant that United will now look forward to the fifth round draw which takes place on Monday evening while the Latics will concentrate on staying in the Championship this season.
Mourinho made nine changes for the game
Jose Mourinho made nine changes to his starting lineup for the game with only Chris Smalling and Marcos Rojo keeping their place in the team for the game.
It showed straight away that their was plenty of changes to the team as the players were very slow from the get go and that gave the visitors the confidence to try and get a goal early on. They almost did just that when Callum Connolly forced Sergio Romero into a very good save from outside the box.
The visitors frustrated the hosts throughout the first half
The Latics were very assured in their play for the first half an hour of the game until United finally started to create some chances.
From their the United players grew in confidence and they should have got the opening goal when they counter attacked on the visitors ending with Timothy Fosu-Mensah producing a wonderful cross to the back post but Mkhitaryan put the ball wide with the goal at his mercy.
Fellaini broke the visitor's resistence on the stroke of half-time
Just as it looked like Wigan were going to get in at half-time level, Fellaini headed home on the stroke of the half-time whistle after meeting a wonderful cross from Schweinsteiger.
The goal was literally the last chance for any team in the first half in which Mourinho will have been happy to see his side in front but the performance wasn't up to the standard United fans expect.
United showed more intent at the start of the second half but they could have conceded an equaliser when Romero made a mistake by spilling a cross to David Perkins but he made up for the mistake by making a good save.
Smalling doubled United's lead before the hour mark
That miss proved costly as United managed to double their lead just before the hour mark when Anthony Martial picked up the ball on the left wing and produced a good cross into the box and Smalling powered a header into the back of the net.
The second goal effectively killed the game as a contest as the visitors started to tire and United started to pick wholes in their defence and that is were the third goal came from.
Two further goals before the end of the game saw comfortable passage for United
The Latics had the ball in a dangerous position but when they lost the ball, the hosts broke at them with pace and the brilliant move was finished off by Mkhitaryan as he turned the ball into an empty net after a good pass from Martial.
Things got even better for Mourinho's side when they managed to add a fourth before the end of the game.
The goal was scored by the forgotten man this season, Schweinsteiger as put the ball into the net from close range after Ander Herrera headed on a Wayne Rooney cross.
It meant that United avoided a cup upset and comfortably made their way through to the fifth round of the FA Cup.