Jesse Lingard’s first United brace secured a routine 2-0 win as the competition holders made safe progress into the quarter-finals.
“We were in the game from the first minutes, pressing and forcing things, the attitude was good,” Jose Mourinho said. “Scott McTominay did very well, strong in the midfield. Axel Tuanzebe started maybe a little shaky, but then we got stability and he came into it.
“There were no injuries either, so it is a good day.”
Victory was just what United needed after the shock of their first Premier League defeat of the season at Huddersfield on Saturday.
But Swansea were accommodating hosts and provided little contradiction to the belief that a long season lies ahead of them.
United boss Jose Mourinho said after victory over Burton in the previous round that they would be better off without this competition on their itinerary, suggesting they would be fresher for Europe.
Mourinho made seven changes for this tie, with only Lingard, Chris Smalling, Ander Herrera, and Anthony Martial remaining from the side shocked on Saturday.
Victor Lindelof had come off the bench there and was culpable for both Huddersfield goals, but the summer signing was given a vote of confidence as he took his place alongside Smalling and teenager Axel Tuanzebe in a three-man defence.
But Lindelof looked bereft of confidence and was embarrassed on more than one occasion by Oli McBurnie, a 21-year-old stand-in who would have been on loan at Barnsley this season but for an administrative error on transfer deadline day.
United started on the front foot with Smalling heading wide from Marcus Rashford’s corner and Lingard forcing Swansea goalkeeper Kristoffer Nordfeldt to tip over his 25-yard effort.
Lingard enjoyed better fortune after 21 minutes when Angel Rangel lost possession on the halfway line and Herrera fizzed a ball forward.
Rashford’s delicious touch laid the ball into the path of Lingard and he slotted the ball low past Nordfeldt with a minimum of fuss.
Swansea’s frustrations grew when Martin Olsson, the only recognised left-back at the club, limped off with a suspected hamstring pull.
But they were spared falling further behind when Scott McTominay lifted his shot over from the edge of the box.
United doubled their lead after 59 minutes from a move as fluent as the one which had produced the first goal.
Tuanzebe strode out of defence purposefully and spread a pass to Matteo Darmian on the right flank.
Darmian’s cross found Lingard between two Swansea defenders and his powerful header from just inside the box gave Nordfeldt no chance.
United should have had a third when Nordfeldt could only palm down McTominay’s header and Tuanzebe blazed over from close range.
Sergio Romero made late stops from substitutes Tammy Abraham and Leroy Fer, but United eased into the last eight after sending on Nemanja Matic and Romelu Lukaku from the bench.
Nordfeldt, Rangel, van der Hoorn, Mawson, Olsson (Naughtonat 39), Ki Sung-yueng, Mesa (Fer 67), Routledge, Clucas, J Ayew, McBurnie (Abraham 72).
Romero, Lindelöf, Smalling, Tuanzebe, Darmian, Herrera (Maticat 67), McTominay, Blind, Lingard, Rashford (Lukakuat 67), Martial (Shawat 87)