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The Argentine struck a double to give the Nerazzurri a slender 2-1 lead after Michele Pazienza had equalised for the visitors.
Icardi needed just six minutes to fire the hosts into the lead by converting a teasing cross from Yuto Nagatomo with a crisp half-volley.
However, Bologna saw their resilience rewarded after Pazienza fired beyond the Slovenian keeper from the edge of the area to draw level.
The Inter starlet hit his second of the evening by unleashing a fierce shot past Curci from outside the box just after the hour mark, only for the pendulum to swing the way of the visitors again as Kone pegged Milan back once more.
There was yet more drama, though, as the hosts were given an opportunity from the penalty spot with six minutes remaining, but Curci was equal to Milito's effort as the sides shared the spoils.