City piled on the pressure in the second half and Bernardo Silva halved the deficit after 69 minutes before Aguero's penalty hit a post and rebounded off goalkeeper Kristoffer Nordfeldt and then City snatched a late victor.
And, just when Swansea looked to have held on to force extra time, Aguero completed the comeback two minutes from the end to send City to Wembley.
Aguero's penalty struck the post and went in off Nordfeldt while Aguero's winning goal, a stooping 88th-minute header, should have been disallowed for offside.
There was also debate over City's equaliser, which came from the penalty spot after Cameron Carter-Vickers was adjudged to have fouled Raheem Sterling.
During his time in Manchester, he'd been loaned out to FC Twente in the Netherland and also Ipswich Town but given the chance to join Swansea, Celina chose to quit City for Swansea after only managing to make four appearances.
Florida State upends Virginia, 69-59, in ACC semifinals
The Cavaliers led briefly during the second half and were down 49-48 after Jerome's 3-pointer off a nasty dribble shake of M.J. In the win over the Cavaliers , Florida State's David Nichols led the team on 14 points with three assists and five rebounds.
"But, at the same time, we played incredibly well, with the amount of actions, the amount of crosses and the goalkeeper (Nordfeldt) was incredible", he added. I don't like to win games when decisions are wrong.
"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.
Those are the two stadiums where Premier League clubs are playing in the quarter-finals of the competition.
"It means a lot because we know how big Wolves were in the 1950s and 1960s and there are people in the stadium who have memories of that".
Police match commander Supt Steve Jones said the safety of supporters, players, officials and those who work at these games was "paramount". But for a long time tonight it looked like that might happen.