Graham Potter’s side gave a fantastic account of themselves at the Liberty Stadium as they went toe-to-toe with the Premier League leaders.
Matt Grimes put them 1-0 up from the penalty spot before Bersant Celina, a former Man City academy player doubled the lead, capping off a sublime team goal.
But City regrouped after the interval after Pep Guardiola’s half-time team talk.
Their quality began to show in the second half and Bernardo Silva pulled one back in the 69th minute.
Swansea then felt incredibly hard done by when Raheem Sterling went down under the challenge of Cameron Carter-Vickers in the penalty box.
Replays showed that the Tottenham loanee appeared to play the ball first, but the referee pointed to the spot.
Sergio Aguero hit the post from the spot kick before it deflected in off the heels goalkeeper Kristoffer Nordfeldt.
The hosts then felt even more aggrieved when the Argentine headed in the winner minutes later, he was standing in an offside position when he met the cross.
With the help of VAR, Swansea might still be in the hat for the semi-final draw but instead they were left to rue their harsh exit.
The FA have decided to operate selective use of the video assistant referee in this season’s competition.
It is only in use in grounds of Premier League clubs. This will only rub salt in the wounds for the Swans, who were a top-flight club last season.
Having only been relegated last year, they will now feel even more bitter about the regulation. With the help of VAR, Swansea may have seen two of the goals they conceded chalked off.
After the match, BT Sport pundit John Hartson made his feelings be known on Andre Marriner’s decision to award a penalty.
He said: ”Sterling, he’s bought that penalty!”
“It shouldn’t have been given, it is NOT a penalty!”
Alan Shearer added: “It is very wrong and very unfair. Swansea’s stadium could have been equipped for VAR, they were a Premier League team not long ago and they will feel hard done by tonight and rightly so.”
City remain on course to secure a memorable quadruple with just over two months until the season’s conclusion.