Milan were hoping to do The Double just days after securing the Scudetto, but were beaten 2-1 on the night, 4-3 on aggregate.
"We had several chances, including one with Robinho in the opening seconds," said the Coach.
"When we did get to the box, we couldn't finish off our opportunities. Palermo played a great game and I can't criticise my players for anything."
The Rossoneri needed a victory to overturn the 2-2 first leg draw at San Siro, but only snatched a late goal with Zlatan Ibrahimovic.
"It was important to keep the game in the balance, knowing that with a few changes we could bring the result home," continued Allegri.
"We went behind, but were also a little unlucky against an excellent Palermo. We did all we could in the circumstances.
"There was a penalty on Ibrahimovic in the final minute, but if the referee didn't give it then there's no point discussing the matter."