In a dramatic clash at the Liberty Stadium, City had gone in at the break with a two-goal deficit against the run of play.
Matt Grimes put Swansea ahead on the 20th minute mark after Fabian Delph brought down Connor Roberts inside the area.
Former City midfielder Bersant Celina grabbed the Welsh side's second of the game shortly after, curling a shot past Ederson and into the top corner of the goal.
However, Bernardo Silva started the fightback by pulling a goal back for the Blues in the 69th minute, making the most of our increasing pressure.
Then the introduction of Sergio Aguero from the bench secured a famous win.
The Argentinian fired home two goals in the final 15 minutes to secure the victory and City's passage through into the FA Cup semi-finals.
Following the dramatic late victory, Bernardo spoke of the incredible come-back, which left City still in contention to achieve an unprecedented quadruple of the Premier League, Champions League, FA Cup and the already secured Carabao Cup.
"After the first goal I think we were perfect. The way we pressed, the way we were with the ball, very aggressive," said the Portuguese midfielder.
"We're very happy with the win. All four competitions, one of them is done, we won. Into the next three."
Asked if he thought the quadruple was possible for City to achieve this season, Aguero added: "Always possible, always.
"Sometimes these comebacks happen, it is important that we've qualified for the semi-finals, so we're happy.
"We always believed it would come, always we fight."
The draw for the semi-finals of the FA Cup will take place immediately after Millwall and Brighton's quarter-final clash on Sunday 17 March.