The 18-year-old Belgium-born winger made his competitive first-team debut in the Community Shield at Wembley as a substitute, Manchester United said on its website.
He scored both goals in United's 2-1 win at Sunderland in the teen's first Barclays Premier League start.
Manchester United manager David Moyes said he was "delighted" about the signing.
"He is an outstanding talent and has impressed me consistently since I came to United in July. His balance, poise and ability to beat players is truly exceptional for a boy of his age. It’s great that he recognizes that this club, with its tradition for developing young players and giving them a chance on the biggest stage, is the best place for him to be," he said.
"It’s a great thrill to be able to sign for Manchester United," Januzaj said. "Since arriving here I have always felt like this was the right club for me. This season has been great for me -- going on the preseason tour, making my debut in the Community Shield win at Wembley and then breaking into the team at Sunderland -- it’s like a dream. I want to work hard now and establish myself as a Manchester United player."