Matteo Lavelli netted the decisive spot-kick for a 3-1 shootout win for the Italian champions as Maresca led out City for the first time since taking over from Pep Guardiola in June.
An open first half saw City take the lead in the 14th minute when Antoine Semenyo jinked towards the byline and crossed for youngster Divin Mubama to tuck in from a few yards out.
Semenyo was a constant menace and almost scored a wonderful solo goal soon afterwards when he weaved through several Inter defenders but could not squeeze the ball past goalkeeper Josep Martinez.
Within minutes the Italian champions equalised when City failed to clear a Jamal Iddrissou cross and the ball fell for Benjamin Pavard to slot home from near the penalty spot.
Both sides missed golden chances soon after the restart but the game began to drift as the substitutions piled up.
City's Vitor Reis sliced a shot against the crossbar while young winger Ryan McAidoo hit the post as the Manchester club pushed for a late winner that never came.
Maresca took over in June and faces a tough task to succeed Guardiola, who won 20 trophies in a glorious decade at City.
Many clubs have struggled after the exit of serial title-winning managers, with local rivals Manchester United notably falling from the pinnacle of the English game after Alex Ferguson retired in 2013.