Genoa, looking for their first victory at Inter in 21 years, struggled from the start.
The home side sent an early warning when Fredy Guarin fed Icardi in the box and his attempt was saved by Genoa goalkeeper Mattia Perin.
Genoa had a good opportunity to take the lead in the seventh minute when Lestienne spotted Luca Antonelli in the box but his left-footed strike was kept out by Inter Milan goalkeeper Samir Handanovic.
Inter went ahead five minutes later when Perin was unable to hold on to Icardi’s powerful shot from 12 yards and former Genoa striker Palacio was first to the rebound, smashing the ball into the back of the net.
In the 35th minute, Perin was forced to fully stretch to clear Palacio’s right-footed shot towards the far post.
Inter continued to press forward and their efforts paid off in the 39th minute when an unmarked Icardi got his head on Hernanes’ corner and his effort went past a helpless Perin.
The visitors responded after the break and Handanovic was twice called into action before the hour mark to deny long-range attempts from Antonelli and Iago Falque.
Handanovic was fortunate when the woodwork denied Lestienne, who drilled a left-footed effort from 12 yards.
Genoa were finally rewarded for their attacking display, though. Tino Costa’s bullet strike was on target and Handanovic was unable to hold on to the ball to leave Izzo with an easy tap-in.
That sparked hope for the visitors, but Mancini breathed a sigh of relief when Vidic’s close-range header from Podolski’s corner beat Perin.