The Women's Library is currently held by London Metropolitan University, but the university decided it could not longer afford to house the collection.
Several leading universities expressed an interest in acquiring the library, which is widely regarded as one of the country's leading specialist archives and attracts around 30,000 visitors a year.
The library will now be passed to the LSE after governors at London Met approved the decision on September.
It follows a high-profile campaign to keep the collection at its current home in Whitechapel, which was backed by celebrities including the broadcaster Sandi Toksvig.
An online petition against London Met's decision to move the library has been signed by almost 12,000 people.
From next year, the collection will move from its current base to become the Women's Library @ LSE, with a new dedicated reading room to ensure the collection's unique identity is maintained. It will also remain open to the public.
Craig Calhoun, director of LSE, said he was "proud that LSE has been able to step in to keep the Women's Library open".
"There are numerous synergies between the Women's Library collection and LSE's existing holdings. Combined, they will undoubtedly make one of the best international collections for the support of research on women's lives and gender issues," he said.
The Women's Library was founded in 1926 as the Library of the London Society for Women's Service, a non-militant organisation led by suffragist Millicent Fawcett.
It contains documents relating to every aspect of women's lives, including women's rights, suffrage, sexuality, health, education, employment, reproductive rights, the family and the home.
The library holds a copy of Mary Wollstonecraft's A Vindication of the Rights of Woman (1792) as well as first editions of the Brontës' works and of Virginia Woolf, which will join items in the LSE's collection such as a first edition of Sylvia Pankhurst's The Home Front (1932) inscribed by the author.
Work will also begin on digitising the Women's Library @ LSE collection and providing access to it through the LSE Digital Library.