The rescued doctor has worked with Morning Star for three years. He serves as its medical adviser, and travels frequently to Afghanistan, the agency said.
Morning Star did not release the identities of the other two men, citing safety concerns.
Joseph is expected to return home to Colorado Springs, Colorado, within a few days, after medical examinations and debriefings, the agency sad.
On Sunday, his family extended condolences to the slain sailor's family.
"We are incredibly grateful for the multiple agencies of the U.S. government that have supported us in this difficult time, and especially the quick response by our military and partner allies to rescue Dilip," the family said. "They showed great heroism and professionalism."
Despite the kidnappings of Joseph and its two other staffers, Morning Star reiterated its "commitment to continue its work" in Afghanistan.