The Rossoneri coach was relieved to end their winless run in Serie A and was delighted with Kaka's contribution, who scored his 99th goal for the club
It was the first time the Rossoneri have put together back-to-back victories since the beginning of September, and eased the pressure on coach Allegri following a spell, which has seen the club go on a five-game winless run in Serie A.
And the Milan coach was relieved that the team came from behind to secure a vital three points, which lifts them into the top half of the table.
"We played a good game with the right intensity," he told reporters. "It was unfortunate to concede the first goal, but we did well not to give up and reverse the result. It was what the club needed after the good win in the Champions League."
Goals from Riccardo Montolivo, Mario Balotelli and Kaka helped seal the victory and Allegri was particularly pleased with the contribution of Kaka, who netted his 99th goal for the club, following up his strike against Celtic in midweek and Stephan El Shaarawy, who continues to recover from injury.
"Kaka is very important to us, he can play on the left on the right and through the middle. El Shaarawy played a good half hour and that will be very important for his recovery," he continued.
The coach also insisted Milan's recent off-pitch turmoil, which has seen the position of the club's vice-president Adriano Galliani come under scrutiny, has not affected himself or his players.
He added: "We have been working hard at Milanello, isolated from everything and everyone. The president took an important decision (to support Galliani) and everyone is working for the good of Milan."