Speaking at a Uniya farewell function on Friday night, Sr Fawkner said she was "both sad and glad to be here".
In March the centre's board publicly announced that it would not be "sustainable" for Uniya to remain open over the long term.
Sr Fawkner characterised Uniya as a "Jesuit voice for justice in a period of the Church's life when many religious congregations were rediscovering that a commitment to the gospel was a commitment to justice and intrinsic to the Christian life".
Recalling a visit to the Uniya office in 1998 or 1999, she described it as a "pretty dingy affair" with founder, Fr Frank Brennan "the only sign of human habitation".
Later becoming the first woman director, Sr Fawkner said she "had the impression, perhaps not only of Uniya, but more generally of the Jesuits, of a group committed to mutuality, to partnership and a genuine openness to work with women".
Paying tribute to founder Fr Brennan, the Good Samaritan sister, who is now on her congregation's leadership team, said that she was impressed by the "Uniya way".
"Working in the field of social justice, the temptation is to go off half-cocked, to take the self-righteous moral high ground, to become even violent in one's rhetoric by playing the blame game, to resort to motherhood statements, or to be enmeshed in ideology. This was not the Uniya way," Sr Fawkner said.
"The Uniya message was always undergirded by the message of the gospel of fullness of life for all, especially for those made poor. But it never retreated into Church speak, or into political theory speak.
"Uniya never spoke down to people, but it did make complex issues accessible. The Uniya way was not to berate or debate, but to respectfully dialogue," she said.
The Jesuit centre is well known for its interventions in a number of public policy issues, including the plight of refugees and asylum seekers, Indigenous rights and foreign policy.
Referring to the Uniya legacy, Sr Fawkner said "it seems to me that within our Church where there should be a multiplicity of voices, because of stretched resources, the diverse voices that are the sign of an adult, vibrant church are dying or mooted, if not overtly being silenced.
"Where else do we have a church-based organisation encouraging forums about the things that really matter to those made poor or marginalised?" Sr Fawkner asked.
"My last impression is that Uniya was never about Uniya. It was never about the Jesuits. It was never about issues or ideology. Uniya was about people, about those disadvantaged or made poor by injustice. The organisation might close but I pray that the Uniya spirit may live on," Sr Fawkner concluded.
Founded by Jesuit public intellectual Fr Brennan, Uniya was named after the first Jesuit mission to Aboriginal people established in 1886 in the Northern Territory's Daly River area.
The Uniya model was built on the experience of earlier Jesuit-created bodies such as the Institute of Social Order and the Asian Bureau of Australia.
Notable former staff include Jesuit Provincial Fr Mark Raper, Professor Tony Vinson and Sydney Bishop Anthony Fisher.
The centre will close its doors for the last time on Thursday.
Sr Patty Fawkner sgs, Uniya farewell (25/5/07)
Fr Frank Brennan, Reflections on my association with Uniya (25/5/07)
Fr Mark Raper, Uniya a meeting place (25/5/07)
LINKS (not necessarily endorsed by Church Resources)
Uniya Jesuit Social Justice Centre
Uniya (old website)
Jesuit Province Australia
Jesuit justice centre "unsustainable" (CathNews, 22/3/07)
Uniya warns over low Australian engagement in Asia-Pacific (CathNews, 7/2/07)
Former diplomat criticises "well-intentioned" human rights advocates (CathNews, 1/8/06)
Jesuit "pastoral cycle" expert visiting Australia (CathNews, 10/5/06)
Research criticises mutual obligation 'franchise' (CathNews, 15/3/06)
Agency says Australia-Malaysia ties must go beyond trade (CathNews, 29/7/05)
Jesuit Centre assesses 'failed state' threat to Australia (CathNews, 8/4/05)
Jesuits scrutinising honesty crisis in public life (CathNews, 21/2/05)
New paper updates awarenesss of Vietnam human rights situation (CathNews, 16/11/04)
Jesuit says 100% of revenue to East Timor would be fair (CathNews, 3/9/04)
Jesuit Centre says Australia must support Indonesia reforms (CathNews, 14/7/04)
Jesuit advocate says there's no good reason for child detention (CathNews, 11/6/04)
Jesuit says dishonest politics making Australia a terror target (CathNews, 16/3/04)
Jesuit group backs Iraqi assertion on Australian treatment of Iraqis (CathNews, 17/4/03)