The BBC understands that 27 of the organisation's 59 posts will move to Extern.
Staff at Fasa, that offers support to people with drug and alcohol addiction, were not paid in March.
A creditors' meeting is due to take place in Belfast later this afternoon.
Fasa announced in March that it was facing "the prospect of immediate and terminal financial insolvency" and was being investigated by the Charity Commission for Northern Ireland.
Before suspending its services, it said it helped more than 2,000 people facing suicide, self-harm, substance abuse and mental health issues at five sites in Northern Ireland.
Fifty-nine people worked for Fasa, while 70 volunteers helped provide its services.
Extern said it would take responsibility for the "harm reduction service", a key Public Health Agency (PHA) funded project held by Fasa "until recent weeks".
The PHA said contracts between the agency and Fasa to provide a range of services had now transferred to Extern.
It said no changes had been made to the contracts and "adults in need of mental health or drug and alcohol support, or young people in need of information, advice and support, will still be able to access and receive the same services as before".
Extern chief executive Charlie Mack said that following "detailed discussions", the Fasa board had "endorsed a proposal from Extern to take on the services provided".
"Whilst Extern had hoped to secure all 59 jobs and service delivery to clients, the complexity of some contracts meant this was not possible," he added.
"However, we are thrilled to be able to deliver this essential contract, secure 27 jobs, and, most important of all ensure continuity of service to clients.
"We remain hopeful that the remaining contracts held by other funders including the Department for Social Development can be considered for procurement as soon as possible to ensure service provision is available to those who are most vulnerable at the point where they need support."
The charity Extern is to take on some of the work carried out by Fasa (Forum for Action on Substance Abuse), which has suspended its services due to financial difficulties.