Sasuke did not killed Itachi,
Sasuke fought Itachi, in what it seem a very close fight. However, as the Manga develops you find out that Itachi was sick and this undermined his fighting ability, however he still controlled most of the fight against his brother.
In the end Itachi dies out of the injuries he sustained and the exhaustion of his body. The exhaustion was mainly because of the use of heavily taxing techniques, which in the end were oriented to make Sasuke use all his chakra, which in turn allowed Orichimaru to manifest himself, and this was Itachi’s plan all along. Itachi sealed Orochimaru with a very special and powerful technique and a special weapon.
In the end Itachi did not died by Sasuke’s hand but he instead sacrificed himself to seal Orochimaru permanently.