The Nerazzurri found themselves in the unusual position of entering the competition at the third-round stage after finishing a lowly ninth in Serie A last season, but they made sure of their place alongside the top-eight seeds by dispatching Cittadella 4-0 at San Siro on Sunday night.
Jonathan scored the first competitive goal of Mazzarri's reign with Rodrigo Palacio netting either side of half-time before Andrea Ranocchia added a fourth goal in the 63rd minute.
Serie B side Cittadella had seen their slim chances of competing crippled after 30 minutes when defender Simone Pecorini earned a red card.
The resolute display was a step in the right direction after one of the worst seasons in Inter's history but new boss Mazzarri knows Genoa will provide a much tougher workout during Sunday's opening league game in Milan.
He told inter.it: "Regardless of the result, it was important for us to produce a convincing display.
"Some of us were a bit tense at the start, there were a few first-game nerves and we haven't reached peak fitness yet but after a few minutes we found a bit of confidence and courage and played some good football.
"The sending-off helped things but it's not the results we need to be looking at in this period; we just need to focus on ourselves and how we can improve. Cittadella were a bit fresher than us."
Four Serie A sides - Cagliari, Genoa, Livorno and Torino - were eliminated by lower-league opposition in the third round.
Inter's reward is a fourth-round meeting with Serie B minnows Trapani and the winners will then visit top-flight outfit Udinese - with a quarter-final clash with rivals AC Milan a possibility for Mazzarri's men.