After a lively start, the Swans went in front on 20 minutes from the spot after Fabian Delph brought down Connor Roberts inside the area. Matt Grimes stepped forward to beat Ederson.
Nine minutes later, the home supporters went into frenzy when Bersant Celina curled home from the edge of the area to double Swansea's lead.
Nordfeldt was at the Swans rescue again with a reaction save off Bernardo Silva's first-time volley on 32 minutes, before David Silva's strike was cleared off the line by Roberts three minutes later.
The Citizens needed to save face in the second half, as Raheem Sterling and Oleksandr Zinchenko replaced Sane and Delph on 57 minutes respectively.
The lead was reduced with 20 minutes remaining, as Bernardo Silva smashed home from the left side after Swansea failed to clear their lines.
The Citizens were awarded a penalty 10 minutes later when Sterling was judged to have been brought down in the box. Aguero took responsibility and his spot-kick deflected in off Nordfeldt for the own goal.
A stunning double save from Gabriel Jesus's header and Aguero's follow-up by Nordfeldt kept City from going in front in the final minutes.
There was no stopping Aguero from turning the match on its head, as the Argentine cushioned sent a header in at the far post with two minutes remaining to seal City's spot in the last four.