The Vatican on Thursday released the document, entitled "I have loved you," which Francis had begun to write in his final months but never finished.
Leo, who was elected in May, credited Francis with the text, cited him repeatedly, but said he had made the document his own and signed it.
The 100-page document traces the history of Christianity's constant concern for poor people, from biblical citations and the teaching of church fathers to the preaching of recent popes about caring for migrants, prisoners and victims of human trafficking.
Leo credits especially women's religious orders with carrying out God's mandate to care for the sick, feed the poor and welcome the stranger, and also praises lay-led popular movements advocating for land, housing and work for the society's most disadvantaged.
The conclusion Leo draws is that the Catholic Church's "preferential option for the poor" has existed from the start, is non-negotiable and is the very essence of what it means to be Christian.
He calls for a renewed commitment to fixing the structural causes of poverty, while providing unquestioning charity to those who need it.
"When the church kneels beside a leper, a malnourished child or an anonymous dying person, she fulfils her deepest vocation: to love the Lord where he is most disfigured," Leo writes.
Leo cites Francis frequently, including in some of the Argentine pope's most-quoted talking points about the global "economy that kills" and criticism of trickle-down economics.
Francis made those points from the very start of his pontificate in 2013, saying he wanted a "church that is poor and for the poor".
"God has a special place in his heart for those who are discriminated against and oppressed, and he asks us, his church, to make a decisive and radical choice in favour of the weakest," Leo writes.
Echoing Francis, Leo rails against the "illusion of happiness" derived from accumulating wealth.
"Thus, in a world where the poor are increasingly numerous, we paradoxically see the growth of a wealthy elite, living in a bubble of comfort and luxury, almost in another world compared to ordinary people," he said.
Francis' frequent criticism of capitalism angered many conservative and wealthy Catholics, especially in the United States, who accused the Argentine Jesuit of being a Marxist.
