The 28-year-old joined the Rossoneri in the summer on a free transfer from Fiorentina and has gone on to form an important part of Massimiliano Allegri's plans at San Siro this season.
And the vice-president believes Montolivo's performance in the 2-1 Serie A victory over Parma on Friday is proof that the midfielder is beginning to fill their former pass master's boots.
"We are putting on an extraordinary chase [for third place in Serie A] and it was important that we won," Galliani told Milan's official website.
"We didn't play in the first half but Montolivo had a good game and is probably playing the best football of his career. We have found another Pirlo."
Galliani also took the time to heap yet more praise on new signing Mario Balotelli following the striker's winning free kick against the Gialloblu.
"I cannot remember such a soft touch to send the ball over the wall [and into the net]," he added. "We all remember Balotelli's powerful efforts but not the soft touches.
"It was [Sulley] Muntari and [Mario] Yepes who went over to [M'Baye] Niang and told him it was Balotelli's free kick which shows what a great team effort it was.
"We are sold out for the Champions League match against Barcelona and it will be a wonderful spectacle."