Replays showed the Argentina striker was marginally offside but the video assistant review system was not in use at the Liberty Stadium, despite having been used in City’s third and fourth-round ties at their Etihad Stadium home.
City had levelled in contentious circumstances too when Cameron Carter-Vickers was judged to have tripped replacement Raheem Sterling in the area and Aguero’s penalty hit the post before rebounding in off Nordfeldt for an own goal.
Defeat was harsh on the impressive hosts, who led 2-0 inside 30 minutes through Matt Grimes’ penalty and Bersant Celina’s spectacular second.
The former City player, who was mocked for an extraordinary penalty miss against West Brom on Wednesday, curled a sumptuous first-time effort beyond Ederson at the end of a tremendous, flowing move.
City fielded a strong side but needed Sterling and Aguero to come off the bench to shake them out of their fatigue, the latter setting up Bernardo for the visitors’ opener, which sparked their thrilling comeback in the last half hour.
Having scored 16 goals in three previous FA Cup games this season and after thrashing Schalke 7-0 in the Champions League on Tuesday, City’s bright start to this game seemed to indicate that this would be another stroll.
Yet after missing a few chances, with Nordfeldt also making two fine saves, the front three of Gabriel Jesus, Leroy Sane and Riyad Mahrez started to lose their composure – and so did City’s defence.
Fabian Delph, who was sent off in his side’s fifth-round defeat by Wigan last year, was caught out by Connor Roberts’ deft touch to bring down a long pass and felled the Swansea right-back from behind to concede a clear penalty for the opener.
The visitors then failed to track the runners as Swansea swept forward, leaving Celina unmarked for the second.
The Premier League leaders carried limited threat in the early stages of the second half and looked on course for a tame exit to a side who are 15th in the Championship.
Yet Sterling and Aguero proved to be the difference after they were brought on either side of the hour mark, the latter calmly picking out Bernardo after his own effort was blocked before the excellent Portugal midfielder curved a fine effort into the far corner.
Sterling then feinted past Carter-Vickers before falling under the challenge. Andre Marriner awarded a penalty but replays suggested the centre-back touched the ball before clipping Sterling’s ankle, and Swansea’s injustice was compounded by Aguero’s fortunate conversion.
Pep Guardiola’s ruthless side were not prepared to settle for extra-time, pushing forward once more and working the ball out to Bernardo on the left of the area to pick out Aguero perfectly, with the lack of VAR working in City’s favour.
“I’m sorry it was offside. I don’t understand why VAR is not used in this competition at this stage. Hopefully next season this won’t happen,” Guardiola told BBC Sport.
“The penalty was really lucky but you need it, especially when you’re in four competitions.”
