Lingard, 24, netted his first ever United brace to ease the pain of their shock defeat away at Huddersfield on Saturday.
The England midfielder gave the Carabao Cup holders a 21st-minute lead at the Liberty Stadium, then doubled it with a thunderous header a minute before the hour to give them a place in the last eight.
United were looking to avoid back-to-back defeats for the first time since September 2016 and boss Jose Mourinho kept faith with five of the losing side from the John Smith's Stadium.
Chris Smalling, Daley Blind, Lingard, Anthony Martial and Marcus Rashford all got a chance to redeem themselves in South Wales.
They responded in the right fashion and the Red Devils could have taken the lead in the opening 90 seconds but Smalling headed wide from Rashford's corner.
The visitors were then denied as Lingard's piledriver from 22 yards was tipped over the bar by keeper Kristoffer Nordfeldt.
But after a decent spell of Swans pressure, United showed the route to goal with an opener of delicious simplicity. Lingard coolly found the corner of the net, placing the ball beyond the reach of Nordfeldt.
It was a lovely strike, coming after Rashford's flick sent him clean through following Ander Herrera's pass.
United had a scare on the halfhour when a cross from Ki Sung Yueng was headed towards his own goal by Blind and hit Oli McBurnie to catapult just wide.
Swansea were dealt a blow when left-back Martin Olsson limped off with a hamstring injury after 39 minutes, handing Kyle Naughton a chance. There was a dramatic end to the first half as United could have doubled their lead while Swansea wasted two half-chances.
Scott McTominay swept a first-time curling effort too high for Manchester United after Herrera and then Martial teed him up.
United made a positive start to the second half and Rashford's wickedly dipping free-kick was punched away by an uncertain Nordfeldt after 52 minutes. But there was no stopping Lingard and he made it 2-0 when he ghosted between Alfie Mawson and Angel Rangel to meet Matteo Darmian's cross with a fierce header and leave Nordfeldt flailing.
Lingard was clearly going for his hat-trick and he wriggled through only to fire across the face of goal soon afterwards.
With United in cruise control, Mourinho brought on Romelu Lukaku and Nemanja Matic for Rashford and Herrera in the 67th minute.