The 17-year-old forward struck in the first half, latching on to Phil Foden's great ball before placing a composed finish beyond the goalkeeper.
Sam Surridge equalised for Bournemouth when he found space to sweep home.
But Foden secured the win late on, stabbing in after Riyad Mahrez's shot hit the post.
Victory means City travel to fellow Premier League side Burnley in the fourth round.
It was a memorable night for Delap, who is the son of former Stoke City midfielder Rory.
The forward was one of five teenagers handed a start by Pep Guardiola in a youthful City side that also included two 20-year-olds.
Delap has only been a part of the City youth team since last year after leaving Derby County but impressed as he led the line, causing problems with his strength and determined runs, and his goal made him City's youngest debutant scorer in seven years, since Rony Lopez against Watford in 2013.
But as impressive as Delap was, Foden was even more so with the 20-year-old the standout player of the match as he dictated play throughout.
Among all Premier League players since the start of last season, only Kevin De Bruyne (eight matches) and Mohamed Salah (six) have both scored and assisted in the same game on more occasions than Foden across all competitions (four).
Bournemouth, relegated to the Championship at the end of last season, gave a good account of themselves, and could have perhaps taken the lead before Foden struck if Surridge had shot sooner when through on goal.
But this performance against a side that has won the competition for the last three seasons in a row will give them heart as they look to mount a challenge for an immediate return to the top flight.
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