The Australian, who won last season's Europa League with Spurs, replaces Nuno at the City Ground after the Portuguese was dismissed late on Monday.
Nuno endured a breakdown in his relationship with Forest owner Evangelos Marinakis, who quickly identified his lead candidate for the role.
"We are bringing a coach to the club who has a proven and consistent record of winning trophies," Marinakis said in a statement.
"His experience of coaching teams at the highest level, along with his desire to build something special with us at Forest, makes him a fantastic person to help us on our journey and achieve consistently all our ambitions.
"After gaining promotion to the Premier League, then building consistently season after season to secure European football, we now must take the right step to compete with the very best and challenge for trophies.
"Ange has the credentials and the track record to do this, and we are excited he is joining us on our ambitious journey."
Postecoglou, 60, is set for a second crack at the Premier League, having left Tottenham despite ending a 17-year wait for a trophy.
Despite winning the Europa League final in May and securing Champions League football for this season, Daniel Levy decided to sack Postecoglou on June 7th, bringing in Thomas Frank from Brentford as his successor.
Postecoglou helped Spurs finish fifth in his first term, but his side struggled with injuries and consistency in a dreadful second league campaign, finishing in a barely believable 17th.
Postecoglou's first game as Forest boss will be away to Arsenal on Saturday, before beginning their Europa League journey against Real Betis later this month.