The Girl Who Takes an Eye for an Eye, will be released worldwide on September 12, publisher Alfred A Knopf said in a statement.
Larsson died in 2004 before the publication of The Girl with the Dragon Tattoo and its two sequels which became worldwide best-sellers with more than 85 million copies sold.
Lagercrantz was commissioned to write a fourth book, The Girl in the Spider's Web, which was published in 2015.
Larsson's books brought the Nordic noir genre of atmospheric, introspective detectives to world attention. The genre includes the "Wallander" novels by the late Henning Mankell as well as the TV shows The Killing and The Bridge.