The triumph sees Milan reach the group phase.
Galliani said: "It is evident this was clearly the most important game of the season. Tonight it was a turning point, gave sense to our great comeback which we earned two minutes from the end against Siena.
"I told the lads we couldn't throw away something that we fought for from January to May. We were all very tense, but we did it and are in the group phase of the Champions League."
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