Larsson died of a heart attack in 2004 at the age of 50 while in the midst of writing a fourth novel about brilliant computer hacker Lisbeth Slander and investigative journalist Mikael Blomqvist.
Lagercrantz's sequel is to be published in August 2015, the BBC said Tuesday.
The first three books in the "Dragon Tattoo" series have sold more than 75 million copies worldwide and been adapted as films.
"We have now decided to let somebody take over and tell what happened next," Eva Gedin from the Swedish publishing house Norstedts told the British broadcaster.
Larsson's partner Eva Gabrielsson said the publisher did not have access to Larsson's manuscript for the fourth novel, so Lagercrantz is starting from scratch, the BBC reported.
Lagercrantz, 51, is a Swedish journalist and the author of several novels, including 2009's "Fall in Wilmslow." He helped soccer player Zlatan Ibrahimovic write his autobiography "I Am Zlatan Ibrahimovic," which was nominated last year for a Swedish literary award, the August Prize.