In 2015 the Labour-run city council agreed the libraries should no longer be funded and managed by the authority as part of budget savings.
Volunteers will staff the buildings, while the books and equipment remain owned by the council.
The council claimed it would help create a "sustainable future" for the city's libraries.
A council report recommended saving six libraries and axing Burgess Road, Cobbett Road, Millbrook, Thornhill and Weston libraries unless community-led initiatives could be set up.
The move was intended to save about Â£285,000 per year.
The proposals faced opposition with an estimated 150 people staging a silent protest through Southampton in March 2015.
The Friends of Cobbett Road Library, along with the newly-formed Burgess Road Library Action Group and Thornhill Community Library group, will staff their respective libraries.
The YMCA will provide library services in Weston, while in Millbrook resources will be moved into an NHS children's centre.
Satvir Kaur, cabinet member for communities, culture and leisure said: "Unlike many other local authorities, despite severe budget cuts, no library buildings will be closing in Southampton."
Five libraries in Southampton under threat of closure are set to be run by charities and community groups.