Delap, the 17-year-old son of former Stoke City favourite Rory, has sparked plenty of excitable talk through his explosive performances for City's youth teams and he wasted little time when it came to making an impression on his senior debut.
Pace, power and a dead-eyed finish in the 18th minute had Pep Guardiola celebrating with delight, but the City boss soon had a furrowed brow as Sam Surridge brought the Championship side level.
Flimsy defending and a failure to create regular clear chances saw Guardiola send for the cavalry in Kevin De Bruyne and Raheem Sterling, with the winner arriving in fortuitous fashion when Foden scuffed a shot into the turf and in after Riyad Mahrez's drive came back off the post.
Foden was one of three players retained from the City side that beat Wolves 3-1 in its Premier League opener and the England midfielder slipped a cute pass to Mahrez in the 16th minute, Bournemouth goalkeeper Mark Travers saving at his near post.
Foden was the provider again two minutes later, with Delap powering past Jack Simpson and clipping a brilliant left-footed finish into the top corner.
City's other debutant, Zack Steffen, was picking the ball out of his net as Bournemouth found a swift, well-worked equaliser - Surridge firing a low shot across the United States international.
The host resumed with renewed purpose, Mahrez going close, but Bournemouth continued to threaten a disjointed defence on the counter-attack.
Surridge twice broke through - a combination of Taylor Harwood-Bellis and Eric Garcia, then a wildly sliding Steffen denying him.
Guardiola had seemingly seen enough and sent on De Bruyne and Sterling in place of holding midfielders Rodri and Tommy Doyle after the hour.
But it was Foden who decisively showed his poaching instincts 15 minutes from time.
Surridge could have sent the tie to penalties but glanced wastefully wide in the 87th minute.