New coach Walter Mazzarri was visibly relieved on the touchline when Rodrigo Palacio sealed victory with Inter's second goal in stoppage time after Yuto Nagatomo had put them ahead with a fortuitous goal.
Mazzarri, Inter's sixth coach since Jose Mourinho left after completing the treble three years ago, was greeted by a lukewarm atmosphere for his first league game since replacing Andrea Stramaccioni who led Inter to a dismal ninth last season.
"These boys hadn't won at home for four and a half months, they had to exorcise a psychosis," Mazzarri told Sky Sports Italia.
"We are getting better and we are finding a balance. We are improving more than I had anticipated, we've done a lot in a short space of time."
Inter dominated the game but, with captain Javier Zanetti, defender Walter Samuel and striker Diego Milito still missing with long-term injuries, their play was disjointed and shoddy.
Genoa, playing their first match under new coach Fabio Liverani, offered little threat of their own and Inter finally broke through with 15 minutes left when Jonathan's deflected cross caught their defence off-guard and Nagatomo headed in at the far post.
Argentine Mauro Icardi, signed from Sampdoria in the close season, saw his header hit the crossbar shortly afterwards before compatriot Palacio latched on to Fredy Guarin's through ball to slip the ball past Mattia Perin.