Wrong.
Orochi is killed by a figured called Susanoo using Kusanagi no Tsurugi, which is was renamed from Ama no Murakumo. This is Kizaru's light blade name, because it's a one of the fabled treasures of Japan.
So Ame no Habakiri, while it alludes to killing Orochi in its name, it does not mean it has to (though I'd love it)
Tldr: Kizaru kills Orochi