Pirlo - winner of three successive Scudetto titles since his arrival in Turin - went on record to blame Allegri for his Milan exit, claiming the Italian boss did not want to play him ahead of Massimo Ambrosini and Mark van Bommel.
But speaking at his unveiling as coach of Juve following Antonio Conte's resignation, Allegri said he was relishing the opportunity to work with the Italy international once again.
"I have always enjoyed a great relationship with him," the 46-year-old said on Wednesday.
"Andrea was, and is still a champion.
"When he was at Milan with me, he went through that season which wasn't particularly lucky for him considering that it was plagued with injuries. But in fact he always played when he was fit.
"He has enjoyed some unforgettable years here at Juve and now I consider myself lucky enough to find him once again.
"Unfortunately in football, sometimes you think that things might go one way, but eventually they take an unexpected turn or head into another direction. And that season, Andrea suffered few injuries and when he came back he struggled to make it to the first team.
"But, as I have explained a thousand times, in the end it just happened. The club, Andrea and I met and he decided to join Juventus and now I am lucky enough to link up with him again after three years. He had a great World Cup, played three fantastic seasons, so I am lucky.
"I've never experienced any problem with Andrea and I can fairly state that I've never called his qualities into question, otherwise people could fairly say that I am a fool."
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