Allegri was appointed on Wednesday following the departure of Antonio Conte, who left Juve by mutual consent 24 hours earlier.
The 46-year-old takes over following a mixed spell with Milan, which saw him win the Serie A title in his first season in charge in 2010-11 before a downturn in fortunes.
Allegri was sacked by Milan in January with the 18-time Italian champions languishing in 11th place in the top flight.
However, he has now been given a chance to restore his reputation with a Juve team that will be looking to win a fourth straight Scudetto in 2014-15.
And, although he was surprised by the offer, the job represents an opportunity Allegri is eager to make the most of.
"For me it's an honour, it was a bolt from the blue when I was called yesterday to ask whether I was available to coach Juventus," Allegri said at a press conference on Wednesday.
"I understand the scepticism of the fans that have seen the coach change in a day.
"I will conquer them with respect and results. I am here to continue and to improve the winning streak."
Juve disappointed in European competition last term as they were knocked out of the UEFA Champions League in the group stages before losing in the semi-finals of the Europa League.
The Turin club will have the chance to bounce back this year, and Allegri was keen to stress the importance of performing well in the Champions League.
"We have a duty to play a great tournament in Europe," he added. "The Champions League has a great charm and Juve certainly deserves to be at least one of the top eight.
"Conte has represented so much for Juve. This team has dominated in Italy but we must improve in the Champions League."
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