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.
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 offguard 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.
(Writing by Brian Homewood in Berne; Editing by John Mehaffey)
