The 17-year-old son of former Stoke midfielder Rory put the holders ahead in Thursday's third-round tie with a fine 18th-minute strike from just inside the box at the Etihad Stadium.
With Sergio Aguero out injured and Gabriel Jesus rested, the youngster grabbed the chance to impress as City, bidding for fourth successive win in the competition and sixth in eight years, booked a fourth-round trip to Burnley.
Delap said: "I was buzzing, obviously, to get the call up – it was a dream come true. To top it with a goal, I can't ask for much more.
"I'll probably be smiling for a couple of weeks."
Despite falling behind to Delap's effort, Bournemouth responded well and levelled within four minutes through Sam Surridge.
The Cherries frustrated City for long spells and manager Pep Guardiola eventually summoned big guns Kevin De Bruyne and Raheem Sterling from the bench.
In the end it took a 75th-minute winner from Phil Foden, tucking home after Riyad Mahrez hit the post, for the hosts to secure victory.
Guardiola was impressed by Delap's first outing.
He said: "We see him in training and he scores a lot. He's a striker we don't have – strong and good in the air.
"We have a lot of confidence with him for the future."
Guardiola made eight changes as he fielded a youthful side.
"It was a young team," he said. "All of them played incredibly. It was good for us. The opponent was tough.
"I felt we deserved the victory. In the second half we were much better."
The downside for City was an injury to 19-year-old Adrian Bernabe, who was carried off in the first half.
"Hopefully it is not serious," said Guardiola, who claimed he now has only 13 fit senior players for this weekend's game against Leicester.
Bournemouth boss Jason Tindall was pleased with the application shown by his side.
He said: "We are disappointed we have gone out of the cup but I'm pleased with the performance.
"We were brave and good in possession and caused Manchester City a few problems.
"It was testament to how we did in the game they brought on the likes of De Bruyne and Sterling.
"I didn't feel they created too many clear-cut opportunities and felt we were a threat. I'm disappointed with losing the game."
Tindall would not be drawn on whether Delap should have been sent off for a challenge on goalkeeper Mark Travers which saw the City youngster booked late in the game.
He said: "I felt it was a late challenge, a bad challenge, a heart-in-mouth-moment to see Trav stay down for some time.
"I was delighted he wasn't hurt but it was difficult to see from where I was in terms of what card it should be."