"That was a dream start for the boy," Hoeness said after the first-round victory. "Still, it's now our job as a club to properly manage the hype that's immediately going to spring up around him."
Stuttgart paid VfL Wolfsburg the full €25million asking fee, without add-ons, to sign the 21-year-old Germany Under-21 international this summer, ending a lengthy pursuit in which sporting director Fabian Wohlgemuth's side had to fend off competition from Porto.
Pejčinović had made clear from early in the process that he wanted the move, with Hoeness confirming the club had "fought for him" throughout the summer.
"It was clear very early on, for him and for us, that we wanted to do this," Hoeness said. "Now he's here."
The striker managed eight league goals in 30 Bundesliga appearances for a Wolfsburg side relegated last season, adding three further goals in two DFB-Pokal matches before his move — form that helped establish him as a fixture with Germany's Under-21 side and pushed his value up significantly despite Wolfsburg's struggles.
His immediate impact in VfB Stuttgart colours has only intensified talk of a step up to full international recognition, something Wohlgemuth is already tracking.
"He will develop. He will continue to establish himself in the Bundesliga," the sporting director said. "And then that path can certainly lead there at some point."
Pejčinović signed a long-term contract at the MHP Arena and arrives to compete alongside Deniz Undav and Ermedin Demirovic in an attacking line Stuttgart have built for a season that includes the Bundesliga, Champions League and DFB-Pokal.
Hoeness's side reached the cup final in each of the past two seasons and remain on course for a third successive appearance after Saturday's win, with Pejčinović now firmly in the mix to fill the void left by Nick Woltemade's late move to Newcastle United last summer.