The hosts swept the ball to Josh Key, who stepped inside his marker to lay off to the Portuguese, who dispatched an unerring effort into the top corner from 20 yards, leaving James Trafford with no chance.
The roof nearly came off the Swansea.com Stadium as the ball round the net, and when Casey cleared the ball from the next City attack that too was cheered to the rafters.
The visitors began to dominate possession, but the 25-minute mark came and went without Fisher having had a save of note to make.
Their best opening came when Cherki skipped into the Swansea area and rolled an inviting ball across the face of the box that Doku could not reach to finish.
The hosts then spurned a massive chance to make it 2-0. Widell read a woefully bad pass from Rico Lewis and was in on goal, but saw Trafford get fingertips to his low left-footed effort.
At the other end, Cherki prodded the ball against a post after Oscar Bobb had tricked his way down the right flank before squaring.
But City did level when Doku’s pace beat the offside trap and he was able to cut inside and find the far net via a fortunate deflection off Swans skipper Cameron Burgess.
Casey cleared a driven Marmoush effort off the line as City continued to press with half-time fast approaching.
Casey intervened again early in the second half, getting across to rob Cherki as he threatened to reach a Marmoush pass after slick build-up.
The Swansea were having to defend manfully, and were relieved to see Divine Mukasa fail to pick out arriving black City shirts at the far post after patient, intricate play from the visitors.
Sheehan sent on Zan Vipotnik and Ronald in place of Idah and Inoussa respectively, while Guardiola called for Josko Gvardiol, John Stones and Phil Foden.
Nico Gonzalez blazed over after a driving run from midfield while Gvardiol headed a corner wide as Swansea reached the final 20 minutes on level terms and tantalizingly close to a quarter-final place.
Fisher then denied the Croatian as he held his awkward shot at the second attempt.
But there was nothing the keeper could do when Marmoush was fed in down the side of the area to thunder an unstoppable rising strike into the roof of the net.
**Swansea City:** Andy Fisher; Josh Key, Kaelan Casey, Cameron Burgess (captain), Ishe Samuels-Smith (Liam Cullen 83); Ethan Galbraith, Goncalo Franco (Jay Fulton 69), Melker Widell (Manuel Benson 69); Zeidane Inoussa (Ronald 59), Josh Tymon; Adam Idah (Zan Vipotnik 59).
**Unused Substitutes:** Paul Farman, Ben Cabango, Jisung Eom, Ricardo Santos.
**Manchester City:** James Trafford, Nathan Ake (captain), Omar Marmoush, Rayan Cherki, Jeremy Doku, Nico Gonzalez, Rayan Ait-Nouri (Josko Gvardiol 63), Abdukodir Khusanov (John Stones 63), Oscar Bobb, Divine Mukasa (Phil Foden 63), Rico Lewis.
**Unused Substitutes:** Marcus Bettinelli, Gianluigi Donnarumma, Savinho, Matheus Nunes, Nico O'Reilly, Kalvin Phillips.
**Referee:** Jarred Gillett
**Attendance:** 20,490