The 17-year-old striker struck a stunning goal to open the scoring before Phil Foden's second half effort sealed the win after the Cherries had levelled.
And Delap, who had spent part of the summer training with City's first team squad, admitted it was the best moment of his fledgling career to date.
"To take my opportunity and score and help us into the next round I couldn't be happier," Delap admitted.
"To make my senior debut and get my first pro goal... it's got to be at the top.
"Playing with the best players in the world - there is no better experience."
"Pep congratulated me on my debut and the goal afterwards so I'm buzzing."
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