The Australian, who left Tottenham Hotspur at the end of last season, takes over from Nuno Espirito Santo.
Sutton commented on BBC 5 Live: "In terms of the way he'll go about it at Nottingham Forest, he's taking over a successful Nottingham Forest team, and the job that Nuno did last season, seventh place in the Premier League.
"It's about building on that and possibly winning a trophy."
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He added: "If you're a Nottingham Forest fan, you might be slightly nervous because of the way that Nuno played and Nottingham Forest fans bought into that, sort of the low to medium block, and then more sort of counter-attack football.
"I think it will be different under Ange.
"They'll play with greater intensity and press higher."
Postecoglou departed Tottenham Hotspur, whom he joined from Celtic in 2023, at the end of last season. He ended the North London club's 17-year trophy drought, leading them to Europa League glory against Manchester United.
Sutton continued: "In regards to the way the play, he did tweak things at the end of last season at Tottenham, it certainly wasn't Angeball.
"It was quite turgid to watch some of the football towards the end of the season under Ange.
"But he does have his principles."
The 60-year-old, who won five domestic trophies out of a possible six during his two-year stint at Celtic, will be keen to deliver silverware to the City Ground.