Allegri also revealed that front pair Zlatan Ibrahimovic and Antonio Cassano would play.
Ibrahimovic has played only once in the last four games due to two different suspensions in the league but he is free to play in the cup.
And having been knocked out of Europe by Tottenham in the second round of the Champions League, Allegri insists his league leaders will not be taking this competition lightly.
"We want to win this trophy. It's our duty to try to reach the Cup final and win the league," he said.
"Tomorrow (Wednesday) I will play my best possible team, Ibra will play with Cassano and probably Pirlo from the first minute."
That will come as good news to the team which will be without injured pair Alessandro Nesta and Pato.
Allegri is expecting a tough test from Delio Rossi's side as he looks to bring Milan a first victory in this competition since 2003.
"Rossi's team have great individuals and quality and they proved that by winning their last match away to Roma," he said.
"We want to honour this competition and the lads really want it because it's been a long time since we won."
Palermo have never won the Italian Cup although they are going in search of their third final appearance.
However, Rossi himself lifted this trophy two seasons ago at the helm of Lazio.
But he knows that his side are the underdogs.
"Milan are the best team in Italy at the moment and they'll want to do their best in front of their own fans," he said.
"For us to get a result, we'll have to give 100 percent."