On a weekend of shocks and surprises, the much changed holders ensured their place in Monday's fifth round draw at the expense of Warren Joyce's Championship strugglers.
Marouane Fellaini headed United in front shortly before the break from Schweinsteiger's cross and Chris Smalling doubled the lead on the hour.
Henrikh Mkhitaryan finished off a superb sweeping move before Basti brought the house down with his acrobatic effort for the fourth late on.
Backed by their vocal travelling support, Wigan began brightly and carved out the first opportunity of the match on twelve minutes.
The wonderfully named Max Power collected the ball and found Callum Connolly, whose well-hit effort was beaten out by stand in stopper Sergio Romero.
In contrast, United were sluggish early on although the returning Luke Shaw showed flashes of his pace and penetration with a couple of bursting runs from deep.
It was from one such foray forward that led to United's first opening when the full back's chipped pass found Juan Mata, but he could only blaze over the bar.
Mkhitaryan dragged a shot wide from close range when he should have done better, but United did not rue that missed and moved ahead from our next attack.
With seconds remaining before the interval, patient build up between Wayne Rooney and Mkhitaryan found Schweinsteiger whose whipped cross was met by Fellaini to emphatically power home his header.
Fosu-Mensah fired wastefully wide but it was the visitors who then should have levelled when Romero saved with legs to keep out David Perkins.
It was a missed chance that came back to haunt the second tier side as United moved 2-0 ahead moments later when Anthony Martial found Smalling to nod in.
Then followed a special moment for highly-rated young defender Axel Tuanzebe who was introduced in place of Fosu-Mensah for his long awaited senior debut.
Romero saved well again, this time from Connolly, but again the Reds launched a breathtaking counter-attack to make the Latics pay.
A superb team move released Martial, who burst into the box and cut the ball back for Mkhitaryan to put the result beyond doubt.
Another debutant was given a debut, Academy keeper Joel Pereira, before Schweinsteiger applied the icing on the cake on 81 minutes.
He acrobatically flicked home substitute Ander Herrera's knockdown from a corner to rubber-stamp a satisfactory day at the office for him and the team.
Our FA Cup defence continues....
Overall team performance: 7/10
United Faithful Man of the Match: Bastian Schweinsteiger