A dry opening half resulted in fewer scoring chances, with each side reduced to shots from range, with Marcus Rashford firing at John Ruddy on 14 minutes.
Sergio Aguero was then in an offside position when he scored the victor in the 88th minute, but the Video Assistant Referee was not in use as the game was not being played at a Premier League ground.
United grabbed a consolation in added time, as Rashford found the bottom corner from poor defending, but Wolves hung on to secure a semi-final berth.
The Swans had led 2-0 against Premier League champions City before Pep Guardiola's side sparked a comeback. "It is not easy".
"We will analyse the game - we deserved to win the game by far - but to win in that way, I'm sorry". "It could have gone either way, Congratulations to Swansea who played very well and made it hard but we kept going, pressing and eventually the goals came".
"He is on top form, he is an exceptional player and he's enjoying those moments that only the world's greatest strikers have which is nice for the team", he said.
Swansea said the club was set up to use VAR given their history of being in the Premier League between 2011 and 2018.
"The penalty was really lucky but you need it, especially when you're in four competitions".
For a long time, the story of the game looked like it would be all about Celina, a player who joined Swansea from Manchester City and someone Guardiola knows all about.
"In the future we could have problems. It's not nice but thankfully nothing happened this time".