Delap, the 17-year-old son of former Stoke midfielder Rory, put the holders ahead in style in their third-round tie at the Etihad Stadium but they were immediately pegged back by Sam Surridge.
Pep Guardiola’s side, who are bidding for a record-equalling eighth League Cup win and a fourth in succession, will now travel to Burnley in the fourth round next week.
Looking ahead to Sunday’s clash with Leicester, City boss Guardiola made eight changes from the side which opened the campaign with victory at Wolves on Monday.
His selections included handing debuts to the impressive striker Delap and American goalkeeper Zack Steffen.
Tommy Doyle and Taylor Harwood-Bellis were also given opportunities but it proved a sad night for 19-year-old Adrian Bernabe, who was carried off injured before half-time.
They then took the lead on 18 minutes as Foden slid Delap through on goal with a fine through-ball.
The teenager took the ball in his stride and swept a left-footed shot into the top corner from just inside the box.
It was a moment to savour for the youngster but City’s lead proved short-lived as the Cherries responded within four minutes.
Surridge was given too much space to turn in the box and buried a low shot past Steffen.
Both sides suffered an injury blow before half-time as Bournemouth’s Lloyd Kelly and then Bernabe were forced off.
Bernabe’s problem looked particularly serious and City boss Pep Guardiola dashed over to console the youngster as he was wheeled past the technical area on a stretcher.
City tried to push after the break but Bournemouth held firm and grew in confidence. Surridge was twice denied as first Harwood-Bellis blocked a shot and then Steffen raced out of his area to cut short another charge.
Guardiola decided to up the tempo just after the hour by summoning De Bruyne and Sterling from the bench.
The pressure on the Bournemouth back line increased and City eventually made their breakthrough.
Mahrez struck the woodwork with a low shot and Foden was in the right place to tuck home on the rebound.
City went in search of more and Delap’s enthusiasm got the better of him as he crashed into Cherries keeper Mark Travers, collecting a booking for the late challenge.
City had a late scare when De Bruyne went down after a blow to the head but he was able to complete the game.