Despite having one eye on the weekend's clash against Tottenham Hotspur, Pep Guardiola named a strong starting line-up that included Raheem Sterling, Gabriel Jesus and Ferran Torres, although Ederson was rested for American goalkeeper Zack Steffen.
The Citizens are yet to face Premier League opposition in the cup this season, having overcome Cheltenham Town and Birmingham City in the previous two rounds.
It took a while for City to get going but they got the opening goal on 30 minutes, as Kyle Walker broke the deadlock with his first goal of the cup campaign.
The second goal arrived shortly after the halftime break as Raheem Sterling opened his FA Cup account for the season with a neat finish.
Jesus got in on the action three minutes later after latching onto Bernardo Silva's cross before smashing the ball home into the back of the net.
The Swans pulled a goal back in the final 13 minutes as Morgan Whittaker fired past Steffen with a delightful finish, but it proved to be nothing more than consolation.
