Club statements on Monday announced Iachini's arrival and thanked the well-travelled Cosmi for his "great professionalism".
Iachini, who has managed Sampdoria, Brescia and Chievo, said he was committed to helping Siena avoid relegation to Serie B.
"I promise to work with enthusiasm to achieve the goal of staying in Serie A," Iachini, who will be officially unveiled to Italian media on Tuesday, told Siena's website.
Siena are four points from safety after 17 of 38 games and play third-placed Napoli on Saturday.
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