The 22-year-old, signed from Palmeiras in August, set up the opening goal for 17-year-old debutant Floyd Samba and then scored his own after the interval, as Manchester City began their defence of the trophy against Norwich at the Etihad on Thursday.
Speaking to Globo Esporte, Allan said: "I'm very happy. I was very anxious for this debut. Not even in my best dreams did I imagine a debut like this.
"So I'm very happy, very happy. It's continuing to work to evolve and be able to help City more and more."
Enzo Maresca made 10 changes from the side that won the Manchester derby against United on Sunday, with Erling Haaland rested and Allan among three players handed a first City start alongside Samba and goalkeeper Gerónimo Rulli.
Allan began on the right, and it was from there that he supplied the cross for Samba, who headed in on 29 minutes after Norwich goalkeeper Daniel Grimshaw flapped at Nico O'Reilly's deep cross and the ball fell to Allan on the opposite flank.
Rayan Cherki made it 2-0 three minutes later after Ayyoub Bouaddi released him. Allan then switched to the left flank after the break and needed only three minutes to make his mark. He jinked past his marker on the edge of the area, cut inside and drove a low left-footed shot into the far corner for his first goal for the club.
Samba added a second and City's fourth before the hour, and substitute Ryan McAidoo completed the 5-0 win late on. The result sends City into a fourth-round home tie against Brighton.
Allan's arrival had been somewhat overshadowed by City's bigger summer moves, including the signing of Bouaddi from Lille. The Brazilian left Palmeiras in a deal worth €40 million, made up of €37.5 million fixed and €2.5 million in bonuses, and signed a five-year contract.
That made him the third-biggest sale in Palmeiras history, behind only Estêvão, who joined Chelsea, and Endrick, who went to Real Madrid.
He had waited a few weeks for his first appearance, but it took just one match for Allan to show why City were prepared to pay that price.