Dyche, 54, becomes Forest's third manager of the season and will take charge of his first match on Thursday when the club hosts Porto in the Europa League.
The former Burnley and Everton manager returns to the City Ground, where he began his playing career as a youth trainee. He will be joined on the coaching staff by former Forest players Ian Woan and Steve Stone.
Postecoglou was dismissed just 17 minutes after Forest's 3-0 defeat to Chelsea on Saturday, having failed to win any of his eight matches in charge. The Australian, who was only appointed in September, lasted just 39 days in the role - the shortest tenure for a permanent manager in Premier League history.
Dyche had been out of work since leaving Everton in January. He brings significant top-flight experience, having managed over 330 Premier League matches. Forest said the new head coach offers 'the perfect blend of character, tactical acumen and proven achievement' and highlighted his teams' trademark defensive organisation and set-piece strength as qualities that suit the current squad.
The recruitment process was led by global head of football Edu Gaspar and global technical director George Syrianos. The club also considered former Manchester City and Italy manager Roberto Mancini, while Fulham's Marco Silva was an admired candidate but deemed difficult to acquire mid-season due to compensation challenges.
Forest are currently 18th in the Premier League with just five points from their opening eight matches, despite significant investment prior to the season exceeding 100 million pounds. The club is also competing in the Europa League this season, having qualified for European competition for the first time in two decades.