Juve moved up to 76 points ahead of second-placed Napoli, who host struggling Hellas Verona on Sunday, while Milan remained sixth on 49, seven points off the European places.
"We could have played better at times," Allegri told Mediaset Premium. "It was a difficult match, because it's never easy to play at the San Siro.
"I have to praise my players, because we took a step
forward today."
Before the match the San Siro crowd paid tribute to club legend Cesare Maldini, who died aged 84 last weekend. Maldini won four Serie A titles and captained the side in 1963 as they became the first Italian team to win the European Cup.
Perhaps buoyed by the sense of occasion, Milan started strongly against Juve, with Mario Balotelli forcing a world-class save from Gianluigi Buffon from a 14th-minute free kick.
Four minutes later, Alex put the hosts in front with a thumping header from Balotelli's corner as Milan responded positively to a punishing week-long training retreat, a consequence of their disappointing recent form.
Juve steadily rediscovered their composure and, moments after Claudio Marchisio stung keeper Gianluigi Donnarumma's finger tips with a powerful effort, Mandzukic equalised from inside the box after being picked out by Alvaro Morata.
DOUBLE SAVE
Buffon kept the scores level in the 51st with an outstanding double save from Carlos Bacca and Balotelli, before Pogba's thunderous free kick struck the post eight minutes later.
"Milan were unlucky to find the excellent Buffon blocking their path," Allegri added.
Pogba marked his excellent performance with the 65th-minute winner, firing his effort into the ground and over Donnarumma as Milan failed to clear their lines.
"The result angers me," Milan coach Sinisa Mihajlovic said. "We didn't deserve to lose this match. I couldn't have asked for more from my team."
Earlier in Frosinone, Inter's Icardi leapt above Leonardo Blanchard in the 74th to head Ivan Perisic's cross past Nicola Leali in a rare moment of quality from the visitors, who provisionally moved above Fiorentina into fourth on 58 points.
Fiore, now two points behind Inter, visit Empoli on Sunday, while third-placed AS Roma occupy the last Champions League spot with 63 points before they host Bologna on Monday.
"We'll give everything we have to overtake Roma but it will be hard," Inter coach Roberto Mancini said. "As long as there are points up for grabs, we must fight."
Frosinone were reduced to 10 men in the 80th minute when Blanchard received a second yellow card following a clumsy tackle on the lively Stevan Jovetic.
The hosts stay one point from the safety zone in 19th place but will be encouraged by a spirited second-half performance with Luca Paganini twice beating Samir Handanovic only for his efforts to rattle off the woodwork.
(Editing by Ken Ferris and Tony Jimenez)
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