Roma lifted the cup for the second year running with a 2-1 victory over the Serie A champions on Saturday night.
"Compliments to Roma for the great match they played," Stankovic told Inter Channel. "We won the Scudetto, which is the most important thing."
He added: "A success achieved with great sacrifice and great effort.
"It's a shame, we really wanted to win the Coppa Italia and you could see that on the pitch.
"We played an excellent match, making Roma suffer especially in the second half.
"All in all I think this has been a positive season. There's a bit of disappointment over the Champions League elimination but we should remember that this team had to do without many players who got injured."
Inter midfielder Patrick Vieira - who won the double with Arsenal twice - was disappointed not to have pulled off the same feat in Italy.
"We wanted to finish with a double, which always has an important sporting significance, but anything can happen in a one-legged match," he told Inter Channel.
"Roma won and it was right to applaud them. They played a great match. We only lacked a bit of luck, otherwise it would have been a draw, a result that could have been right."