Inter, who finished a dismal ninth last season, toiled without reward for 75 minutes against Genoa until Yuto Nagatomo's fortuitous finish sparked a late rally.
Rodrigo Palacio's goal in second-half stoppage time flattered Inter as Genoa, who narrowly avoided relegation in May, had looked a much-improved side across 90 minutes.
Mazzarri's switch to a 3-5-1-1 formation allowed Inter to dominate possession but lone striker Palacio often cut an isolated figure.
Andrea Bertolacci skipped past a Nerazzurro shirt in the early exchanges, Hugo Campagnaro shutting him down, but there was no one watching Alberto Gilardino when he missed with a free header.
Fredy Guarin then fed Palacio with a sublime through-ball but Thomas Manfredini cleared before he could pull the trigger.
Andrea Ranocchia came closer for Inter when nodding Ricky Alvarez's cross wide and, in the 26th minute, Guarin pulled up just short of Jonathan's low cross to the far post.
Genoa broke forward just before half-time and Juraj Kucka glanced Francesco Lodi's cross wide before the referee blew his whistle.
Samir Handanovic needed to snatch the ball from Bertolacci's feet after he pounced on Gilardino's lay-off following the restart.
Mazzarri introduced former Sampdoria striker Mauro Icardi on 53 minutes and the move almost paid immediate dividends with the tricky Argentinian producing the first shot on target.
His free header from a Campagnaro pass was easily stopped by Mattia Perin, but the chance sparked much needed urgency in both teams.
Kucka bamboozled Jonathan to drive the ball into the Inter box, Ranocchia clearing, and Guarin soon raced up the other end to feed Palacio, who was a yard shy of tucking home.
The near miss boosted the hosts' confidence and, after Perin had blocked a snapshot from Guarin, Inter finally found a way through.
It was not immediately clear who had scored as Jonathan's cross bounced off Luca Antonelli and a surprised Perin, before clattering off a grateful Nagatomo on the goal-line.
Inter wanted a second goal to kill the game off and Icardi soon hit the crossbar with a header before Perin denied Ranocchia with a fingertip save.
The late siege would wear down Genoa's resolve and a splendid assist from Guarin three minutes into time added on allowed Palacio free rein to clip home over Perin's shoulders.