Allegri was sacked by Milan midway through last season as the club toiled in mid-table.
Juventus, on the other hand, romped to a third consecutive Serie A title, under the leadership of former player Antonio Conte.
Yet Conte surprisingly departed in the off-season and was subsequently replaced by Allegri, who began his tenure with a 1-0 win at Chievo on the opening day of the 2014-15 season.
Boniperti, who spent 15 years at Juve between 1946 and 1961 and won five Scudetti, has expressed his confidence in the abilities of the new coach.
"I know his history and he seems like a good guy," Boniperti told Tuttosport.
"He is a coach who is up to the job. However, for Allegri, like for everyone else, what will count are the results.
"Juve's destiny is always the same, every season. They start each campaign to compete for every objective at home and abroad.
"The new signings? It is not my style to judge, but Juventus are a club who must live up to being Juventus."