England international Marcus Rashford netted just before the break with a dinked finish – far from what the Green and Whites deserved – and Ander Herrera doubled their advantage on 61 minutes.
Jesse Lingard came off the bench to add a late third and Romelu Lukaku put gloss on the away side’s victory to see the Premier League giants progress.
Darren Way made two changes from the Sky Bet League Two defeat to Chesterfield.
Suspended former Manchester United trainee Otis Khan missed out and in came the goalscorer of last week Lewis Wing, while Omar Sowunmi returned to partner Nathan Smith at centre-back.
There was a place on the bench for new recruit Alex Fisher who joined from Motherwell just hours before kick-off.
Yeovil were in United’s faces from the off, Francois Zoko going into the book after 30 seconds for a ‘let him know you’re there’ challenge.
The visitors’ new recruit Alexis Sanchez was booed with every touch but it was the Glovers wide man Jordan Green who stole in for the game’s opening chance.
The pacey winger sped past Matteo Darmian and forced Sergio Romero into a neat save with his legs.
Town looked threatening on both flanks, Jake Gray causing a nuisance and Tom James’ teasing ball across the corridor of uncertainty almost tapped in by Sam Surridge.
Romero was called into action again soon after, comfortably claiming Omar Sowunmi’s header from a wicked Gray delivery.
At the other end, Artur Krysiak produced an excellent fingertip stop to nudge a Scott McTominay goalbound effort around the post in United’s first meaningful venture into the hosts’ area.
A rapturous sarcastic cheer met a Sanchez free kick which Krysiak gathered with ease before ‘what a waste of money’ rang around Huish Park following an uncharacteristic wayward pass from the Chile international.
Then James saw a dipping set-piece punched clear by Romero and Sanchez dragged a shot wide before United ahead against the run of play.
There didn’t seem to be an opportunity in the offing but following indecision in the Yeovil defensive ranks, Rashford creeped in to lift the ball over Krysiak into the back of the net.
The goal seemed to settle Jose Mourinho’s side who nearly added another immediately after the interval; Rashford’s poke was deflected behind and Juan Mata hit the bar with a cross-cum-shot.
United showed their Premier League class in the second period and added a second just past the hour.
From a Yeovil attack, the Red Devils hit their fourth tier counterparts with a swift counter-attack which was finished off emphatically by Herrera.
Zoko’s acrobatics warmed Romero’s gloves before Fisher was introduced to the fray for his debut – not a bad side to make your first outing for your new club against.
Krysiak saved tidily to deny Rashford and Angel Gomes found the sidenetting before Lingard and Lukaku wrapped the game in the closing stages.
Town fought until the very last whistle and Way’s men can be very proud of their brave display but quality told in the end.
Yeovil Town: Krysiak, James, Sowunmi, N.Smith, Dickson, Gray, Wing, Bird (Browne 56), Green (Fisher 69), Surridge, Zoko.
Bookings: Zoko 1, N.Smith 29, Sowunmi 47
Not Used: Maddsion, Whelan, C.Smith, Gobern, Santos.
Manchester United: Romero, Darmian, Lindelof, Rojo, Shaw, McTominay, Herrera, Carrick, Rashford, Mata (Lukaku 65), Sanchez (Lingard 72).
Bookings: McTominay 81.
Not Used: Pereira, Smalling, Gomes, Matic, Young.