Pep Guardiola named a young a starting line-up, which included a debut for 17-year-old Liam Delap.
And the son of former Stoke City midfielder Rory didn't disappoint after scoring the opening goal on the 18th minute mark.
Sam Surridge brought things level only four minutes later for the recently-relegated Cherries.
But Phil Foden was able to beat Mark Travers with fifteen minutes left to book City's passage to the next round.
The victory means City have not lost an EFL Cup game since in four years.
"Since I was a boy all I dreamed of was playing senior football," Delap said after the final whistle.
"To make my debut tonight and get a goal, there is no better feeling to be honest."