Kaka provided the assist for compatriot Robinho to put Milan ahead before Lionel Messi earned a point for the Spanish giants at the San Siro.
"Kaka has played an extraordinary game," Allegri said to Gazzetta dello Sport. "He has proved what a champion he is.
"At certain times during the game he played in a different role and he did well.
"To be a champion, you not only have to prove it with your feet but also with your head."
Kaka, who re-joined Milan this summer after four seasons at Real Madrid, only returned to action last weekend after being sidelined with a hamstring injury for a month.
"It's never easy to play against Barca but I thought we did well against them," Kaka said.
"We are growing as a team and I think this Milan can become a strong group.
"I think we have the quality to do well. Individuals may win games but you need to play as a team in order to win titles."
Allegri was pleased with his team's improvement in defence after the Rossoneri limited Barca to six shots on target.
"My players did a good job," he said. "They gave very little away and we defended deeply.
"We also had our chances to win but, overall, I am pleased with the result."
The result has left Milan second in Group H, two points behind pool leaders Barca and two points clear of Celtic.
Milan visit Barcelona on November 6 before they take on Celtic in Glasgow on November 26.
The Italians, who are unbeaten in the competition so far, end the group stage at home to Ajax on December 11.
"We will try to get a result at the Camp Nou," Allegri said.
"But the important games for us will be at Celtic and against Ajax."