The hosts took the lead in the fourth minute when Kovacic's strike deflected into the path of Icardi and, after seeing his initial effort saved by the keeper, the striker followed up to slot the ball home.
Sassuolo came close to a response moments later as Francesco Magnanelli fired in a strike from range that went just over the top corner.
In the 21st minute, the hosts grabbed their second goal of the match through Kovacic, who ran onto Osvaldo's pass and slotted an effort past the keeper.
On the half-hour mark, Inter made it three when Icardi picked up the ball just outside the box before bending a low strike into the bottom corner.
With two minutes of the first half left to play, the hosts further extended their lead when Osvaldo tapped home Dodo's pull back from close range.