The striker, one of five teenagers to start in a much-changed and youthful City side, put the hosts ahead with a powerful drive.
Bournemouth levelled after a smart turn and shot from Sam Surridge beat our other debutant Zack Steffen, but with a penalty shootout looming large, the impressive Phil Foden sent City through with a tap in after Riyad Mahrez's effort came back off the post.
It proved to be a case of third time lucky for Foden, as he clipped a cross field ball invitingly into Delap’s path and the debutant gleefully accepted it as he burst into the area and fired into the top corner off his left foot.
Bournemouth equalised within four minutes following a surging run down the left from former City Academy player, David Brooks.
Dan Gosling collected his pass and poked the ball though for Surridge, who swivelled away from Adrian Bernabe before picking out the bottom corner with a shot across goal.
The Cherries were more aggressive in the second period and Surridge might have extended their lead on two occasions.
Clear cut chances were proving few and far between as the prospect of a penalty shootout became increasingly likely, but City made the crucial breakthrough with 15 minutes remaining.
Mahrez found a yard of space as he advanced to the edge of the area, beating Travers with a low shot that bounced invitingly off the post and into Foden’s path, who tapped into the unguarded net from eight yards.
The kids are alright
Foden, Garcia and Torres would all expect to be involved on night’s like this, but for Bernabe, Doyle, Harwood-Bellis and goalscorer Delap, Guardioa’s decision to make eight changes was a real opportunity.
It was a close encounter and one Bournemouth will feel they could have taken to penalties but for Surridge’s hesitancy, but City’s youngsters more than held their own.
Doyle, in particular, showed a maturity beyond his years to get on the ball and dictate play from deep.
Chances were few and far between for Delap, who is not 18 until February, but he took his goal like a season striker, whilst his physicality and pace ensured he did not look out of place in the senior environment.
Man of the Match - Phil Foden
A goal and assist for the midfielder who produced an accomplished display as one of the more experienced players in a young team.
Glided into pockets of space where he could worry the Cherries' defence and did so on several occasions with some incisive passing.
what it means
City progress to the fourth round of the competition, where we'll face Burnley at Turf Moor. The Clarets beat Sheffield United on penalties in round three, winning 5-4 in the shootout after a 1-1 draw.
It's a quick turn around for Guardiola's men, who welcome unbeaten Premier League leaders Leicester City to the Etihad on Sunday 27 September. Kick-off is at 16:30 (UK)
BOURNEMOUTH: Travers, Smith, Ofoborh, Simpson, Kelly (Zemura 28'), Rico (S. Cook 63'), Gosling, Billing, L. Cook, Brooks, Surridge.
SUBS: Begovic, Stacey, Lerma, Kilkenny, Anthony.