The Nerazzurri had not won at the San Siro since Tommasi Rocchi scored to defeat Parma in April.
"These lads had not won at home for many months and suffered so many defeats here last year, so they needed this win to exorcise the psychosis," Mazzarri told Sky Sport Italia.
"I tried to reassure the squad, as I knew Genoa would sit and wait for us. I am happy for the lads, also for their first half performance. We allowed Genoa very few scoring opportunities, so that means our team is growing.
"The wingers did very well and are following my instructions. They are still learning and when there are spaces we have to move quicker. This team is improving, perhaps quicker than I expected. Once we broke through, we played good football."
Inter have to wait until September 15 before playing at the San Siro again, when they entertain reigning champions Juventus.