Head coach Sinisa Mihajlovic was under severe pressure at San Siro as the club struggled for consistency in the first half of the season, but they sealed a 6-0 aggregate victory over Alessandria on Tuesday in the Coppa Italia.
That meant Milan secured a place in the final for the first time in 13 years and are now unbeaten in 12 matches in all competitions, a run that has also seen them rekindle hopes of earning Champions League spot in Serie A.
The turnaround has thrilled experienced campaigner Abbiati, who deputised for teenage sensation Gianluigi Donnarumma in midweek.
"Reaching the final was one of our objectives this season and we did it," the 38-year-old told Rai .
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"We have always believed, despite our initial difficulties. We are trying to come back and it is not easy, but we are doing well.
"We are very happy to have made the final and now we are thinking about the league, which is very important for us – Sunday's game [away to Sassuolo] will be very difficult for us."
Milan sit sixth in the table with 11 games remaining, six points behind Roma in third spot.