Lingard, who hasn't been a regular starter for United this season, showed his quality after giving the visitors the lead after a good passing moved opened up the hosts'.
The winger then added a second with a brilliant header from just inside the box just before the hour mark as the visitors eased past Swansea, who really struggled all night to test the visitors.
Both managers made plenty of changes for the game
As expected before the game, both Paul Clement and Jose Mourinho made plenty of changes to their starting lineups for the game with the visitors employing a 3-5-2 formation with Axel Tuanzebe making his first start of the season.
United's strong start to the game rewarded with Lingard's well-taken goal
The visitors knew that they owed their manager a performance after their disappointing result at the weekend and they started really positively as firstly Chris Smalling put a header just wide of the post, before Kristoffer Nordfeldt denied Lingard with a good save after the winger hit a great shot towards goal from the edge of the box.
United though did score the opening goal of the game 21 minutes in when a lovely flick from Marcus Rashford from a good Ander Herrera pass, found Lingard in on goal and he confidently curled the ball into the back of the net.
The hosts' threatened but didn't really test Romero
After going into the lead though, United took their foot off the pedal and that allowed the hosts' to come more into the game and but for some last ditch tackles from the United defenders, they would have scored an equaliser.
They didn't though and at the break United had a one goal lead just as they had in the league meeting between the sides earlier in the season and they would have been confident of going on to score more, while the hosts' showed glimpses towards the end of the first half.
A brilliant Lingard header put United two goals in front
The second half started very scrappily as both sides struggled to take control of the ball until a moment of magic from Lingard put United into a two-goal lead.
The goal started after a positive run from Tuanzebe ended with a good cross from Matteo Darmian being headed into the back of the net from Lingard giving Nordfeldt with no chance of saving it.
It should have been three soon after as Tuanzebe somehow put his shot over the bar after Nordfeldt saved a header from Smalling from a corner kick.
Romero's brilliance ensures that United keep a clean sheet and progress to the next round
They didn't make it three though and that gave the hosts' some hope as finally they tested Sergio Romero in goal as good shot from Tammy Abraham, on as a second half substitute, was saved by the Argentinean.
At the other end, Lingard almost completed his hattrick when he picked the ball up 30 yards from goal before unleashing a powerful shot which Nordfeldt done well to save.
The only thing missing for United on the night was a clean sheet and they were very thankful for a brilliant reaction save from Romero in injury time after Leroy Fer volleyed towards goal from a corner kick from Ki Sung-yueng.
That was the last action of the game though as United progressed to the next round of the competition, while the hosts' will now put all of their focus on the Premier League as they try and climb up the table to get out of trouble.