The transfer fee was undisclosed but British media reports have estimated it to be around £36 million.
Having been called up by France for friendlies against Portugal in Lisbon and Serbia in Bordeaux, Martial was allowed to leave the training camp to sign his contract on deadline day.
The clubs announced the deal less than an hour before the Premier League transfer window closed.
The 19-year-old scored nine goals in 35 Ligue 1 appearances last season.
"I am so excited to be joining Manchester United. I have enjoyed my time at AS Monaco and I would like to thank them and the fans for everything they have done for me," Martial said in a statement on the club website.
"I have always wanted to play in the Premier League and to join the biggest club in the world is what every young footballer dreams of. I am looking forward to meeting my new team mates and working with (manager) Louis van Gaal who has achieved so much in his career."
Van Gaal said: "Anthony is a naturally talented, young, multi-functional forward with great potential. We have been watching him for a while now and he has developed immensely during his time at AS Monaco.
"I am delighted he has joined Manchester United as I believe this is the right club for him to continue his development as a young player.
"He has all the attributes to become a top football player. However, we need to give him time to adjust to his new environment and the rhythm of the Premier League."
