Pep Guardiola named a strong starting line-up for a clash with Championship opposition.
But the Spaniard saw that selection fall two goals down inside 29 minutes.
A shock appeared to be on the cards, with the fabled magic of the FA Cup seemingly in the air at the Liberty Stadium.
City have, however, shown that they will not be surrendering any trophy without a fight this season.
A stunning second-half comeback was sparked by Bernardo Silva in the 69th minute as he offered the Blues hope.
Fortune favoured Guardiola’s side from that point, with a penalty call going their way after Raheem Sterling took a tumble under minimal contact from Cameron Carter-Vickers – who appeared to get the ball.
Sergio Aguero’s subsequent spot-kick hit the post, but bounced into the net off the boots of unfortunate Swans goalkeeper Kristoffer Nordfeldt.
The Argentine striker than completed a stunning turnaround two minutes from time when he nodded home what proved to be the winner.
That effort came from an offside position, but the match officials spotted no infringement and had no technological assistance to call upon.
VAR was available in South Wales, but the Football Association took the decision not to use it.
That call has not gone down well, with City considered to have been the beneficiaries of goals that should never have stood while brave Swansea are forced to bow out wondering what could have been.
Re Aguero offside winner, the FA took decision not to have VAR at Swansea despite the stadium being equipped for it. The argument was wanted to remain consistent that only PL stadiums would have VAR.— Martyn Ziegler (@martynziegler) March 16, 2019
A penalty that wasn’t a penalty & an offside goal for Manchester City. VAR, where you at?— Ryan. 🔴 (@Vintage_Utd) March 16, 2019
That's why its wrong that VAR is only available at select FA Cup games. Not a level playing field— Nicholas (@jabbathesutt) March 16, 2019
Incredible fight back from Man City. But shocking that there’s VAR at some games and not others this late in the competition.— John Cross (@johncrossmirror) March 16, 2019
Pretty ridiculous that you get to the quarter final stage of the FA Cup, but not all games have VAR.— Mark Ogden (@MarkOgden_) March 16, 2019
Swansea 2-3 Man City - got to have VAR in all games or none. Pen was never a pen and winner was offside— Ian Abrahams (Moose) (@BroadcastMoose) March 16, 2019
Manchester City needed a dive from Raheem Sterling, a dodgy penalty and an offside goal to win against Swansea City. The officiating in English football is hilarious.— FootballFunnys (@FootballFunnnys) March 16, 2019