Allegri was chosen to replace former head coach Antonio Conte, who surprisingly tendered his resignation in July.
Under Conte - who has since been named Italy boss - Juve won three straight Serie A titles, but failed to make an impression in Europe's premier competition.
Midfielder Marchisio hailed the style of both coaches, but feels Juve will be able to improve in the Champions League under Allegri.
"With Conte everything was more planned and the results speak for themselves - three league titles in a row and two Super Cups," he told Tuttosport.
"Now it is different, Allegri has a different idea of the game, as we saw against Milan.
"We work mainly on the possession of the ball. In Europe it works well, we saw it again last year when we faced teams - they played safe football, whilst we had some difficulties.
"We often speak of our failure in Europe, but the first year [2012-13 season] we went out against Bayern [Munich] and last time we threw away qualification against Galatasaray."
Conte's exit has led to suggestions that Juve's domestic dominance will end, but Marchisio sees no reason why that should be the case.
"It is wrong to say this is the end of a cycle. It is Juve who continues - our nature compels us to win again," he added.
"To win a fourth Scudetto is difficult, but we want to succeed. There are many rivals - Roma [are] ahead of the rest, but I also see Inter, Milan, Napoli and also Fiorentina [as challengers]."
