City won 2-1 thanks to first-half goals by Jeremy Doku and Matheus Nunes at Etihad Stadium as the third round continued on Tuesday with fellow top-flight teams Chelsea, Aston Villa and Leicester also advancing.
Chelsea thrashed fourth-division Barrow 5-0, with Christopher Nkunku scoring a hat trick, while Aston Villa won 2-1 at third-tier Wycombe and Leicester triumphed 3-0 in a penalty shootout after a 0-0 draw at fourth-tier Walsall.
The last 32 of the competition is being spread over two midweeks to accommodate the expanded Champions League and Europa League.
Manchester United and Tottenham were among those to advance last week, while Liverpool and Arsenal play their games on Wednesday.
Doku was one of just two Man City players — along with Kyle Walker — to start both the dramatic 2-2 draw with Arsenal on Sunday and against Watford.
The Belgium winger put his team ahead in the fifth minute by shooting home from just inside the area after City seized on a misplaced back-pass.
Nunes added the second in 38th with a low, left-footed shot. Tom Ince scored for Watford in the 86th.
Manchester City boss Pep Guardiola indicated the League Cup was low among his priorities.
"We're not going to waste energy for this competition for sure," he said.
"If I have a lot of injuries, I'm going to play the second team, but we will have all the squad ready and it's a good competition (for some players) to take the rhythm, that's for sure."
Chelsea also rotated its line-up, making 11 changes from the team that started the 3-0 win at West Ham on Saturday.
Nkunku scored the opening two goals by the 15th minute and rounded off the scoring in the 75th. In between was an own-goal and a tap-in by Pedro Neto for his first goal for Chelsea.
Colombia forward Jhon Duran continued his brilliant start to the season for Villa by scoring the team's second goal — from the penalty spot in the 85th — to add to Emi Buendia's opener in the 55th.
The 20-year-old Duran has netted four times in five league matches so far this season, all off the bench.
Wycombe's consolation came in deep into stoppage time.
Walsall were unable to complete an upset over Leicester after taking its Premier League opponent to spot kicks at home.
Leicester converted all three of its penalties, while Walsall failed with its first three attempts to bring the shootout to its earliest possible conclusion.
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