Victory for the Rossoneri was far from smooth sailing, as Catania raced into an early lead after Nicola Legrottaglie rose highest to head home from a Francesco Lodi corner.
The turning point of the game came four minutes after the break when the Rossazzurri were reduced to 10 men following Pablo Barrientos' sending off for a second yellow card.
Milan then took full advantage of their numerical superiority to hit back with a brace from in-form striker Stephan El Shaarawy and a well taken strike from Kevin-Prince Boateng as the Rossoneri picked up their third consecutive win in all competitions.
Despite a shaky start, Allegri was pleased with the way his side's performance against a strong Catania side, although he conceded Barrientos' dismissal was a key point of the match.
"At the start it was all us then they did well to get the goal from the corner and they played aggressively after that," Allegri was quoted as saying by the official Milan website.
"Catania did well when it was 11 against 11 and the Borrientos sending off changed the game.
"We knew Catania would do well. They played good games against Lazio and Juventus and we knew Lodi hits a good free kick.
"Overall we did well in possession and created a lot of chances."
The result moves Milan up two places to seventh in the table, and Allegri believes his men have finally put their early season woes, which saw them pick up a meagre 11 points from their first 10 games, firmly behind them.
"We're a team that has been completely changed [over the summer]," the 45-year-old coach explained.
"Players who made history with the club have left.
"We were a lot younger than the other team here tonight [Friday] and you need time for the team to grow.
"It's the first time we won two in a row and now we have a good position in the league, we'll see where we are in February.
"Coming back from behind is a good sign; it means that physically the team is in good shape.
"We always play better in the second half than in the first.
"We're improving and getting better in the kind of situations that penalised us in the past."
Milan's next game sees them take on Russian outfit Zenit St Petersburg in the Champions League on Tuesday.