The offer of financial support was "amazing", he said.
"To me, it's a win-win. I've never had that in my life."
Dooley said yesterday his contribution was "personal support" and did not come from diocesan funds.
The offer came after a series of meetings between the pair, and because it was "the right thing to do", he said.
Pope Francis had encouraged bishops to prepare for next year's gathering by talking to survivors, and it was important for people like Smith to attend as well, Dooley believed.
"With things like that, you need the survivors there. They're the ones who are the important people to listen to."
Smith was sexually abused in New Zealand and Australia while in the care of the Catholic order, the Brothers Hospitallers of St John of God, beginning in the late 1970s.
He had already written to the Pontiff's Office seeking an audience with Pope Francis, either individually or with a group.
Dooley was also preparing a letter of support for Smith, and believed an audience with the Pope was a possibility.
"I would not be surprised if there's some sort of opportunity — the Pope is keen to listen to survivors, and if that was the case then I would certainly support Darryl in asking."