Just a day after securing his £65 million ($87 million) move from Bournemouth, Semenyo was among the scorers as City put League One Exeter to the sword at the Etihad.
Erling Haaland started but was remarkably not among the goals before being withdrawn at half-time.
Rico Lewis scored twice, while Max Alleyne, Rodri, Rico Lewis, Tijjani Reijnders, Nico O'Riley and Ryan McAidoo were also on target for City and Exeter netted two own goals.
Sunderland won another of the all Premier League ties on penalties against Everton after a 1-1 draw.
Robin Roefs saved all three of the Toffees spot-kicks in the shootout to send the Black Cats into round four.
Premier League strugglers Wolves are now unbeaten in four games after thrashing Shrewsbury 6-1 thanks to a Jorgen Strand Larsen hat-trick.
Fulham needed three goals in the final half hour to beat Middlesbrough 3-1 at Craven Cottage, but Burnley found Championship opposition an easier proposition in thrashing Millwall 5-1.
Brentford cruised to a 2-0 win at financially stricken Sheffield Wednesday.
Newcastle edged past top-flight rivals Bournemouth 7-6 on penalties following a thrilling 3-3 draw after extra time. Newcastle forward Anthony Gordon made it 2-2 with a 95th-minute penalty, and after Harvey Barnes thought he had won it for the hosts in the 118th minute, Marcus Tavernier struck in stoppage time at the end of extra time to set up a shootout.
Pressure mounting on Frank after Spurs loss
Aston Villa heaped more pressure on Tottenham Hotspur manager Thomas Frank after a 2-1 win in London.
Emiliano Buendia and Morgan Rogers put Villa in control by half-time, but Tottenham were vastly improved after the break, and Wilson Odobert's reply set up an absorbing finale.
Spurs were unable to find a leveller, which saw Frank face a volley of boos after the final whistle.
