Pep Guardiola's men looked to be heading out of the competition at the hands of second-tier opposition, but three goals in the final 21 minutes saw City come back from the brink, as Aguero – who appeared to be offside – nodded in the late clincher.
Although the Premier League champions initially looked bright, a confident Matt Grimes penalty put Swansea in the ascendancy and they doubled their advantage with an exceptionally worked Bersant Celina goal against his former club.
The visitors upped the ante after the break and pulled one back through Bernardo Silva, before an Aguero penalty went in off goalkeeper Kristoffer Nordfeldt.
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.
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