Voting at the bi-annual Conference of French Bishops (CEF) in the southern city of Lourdes, a large majority of the 120 bishops approved the payments to those who had suffered abuse within the Church.
The size of the payouts will be made will be determined at a meeting in April, conference chairman Eric de Moulins-Beaufort, the archbishop of Reims, told reporters.
But any money paid was designed neither as compensation that would be determined by a court "or by canonical justice", nor as reparation.
But some victims said this did not go far enough.
"The word 'responsibility' of the Church does not appear, that really bothers me," said Jean-Luc Souveton, a priest and a member of the working group on the issue, who was himself abused by a priest when a child.
Michel, another priest who was also a victim but who did not want to give his full name, agreed with Souveton that the statement was not enough.
But De Moulins-Beaufort, who is Archbishop of Reims, did acknowledge the "silence, negligence, indifference, an absence of reaction, bad decisions or dysfunctionality at the heart of the Church".
An independent commission set up by the Church to investigate the scandal started work in June.
Committee chairman Jean-Marc Sauve told AFP in September that they had received about 2,000 messages in its first three months.
Most of those who had come forward were older then 50, and two-thirds were men, he added. The committee is looking at allegations dating as far back as the 1950s.
On Saturday, another victim of abuse, Olivier Savignac, objected that the bishops had not waited for the findings of the independent commission.
"The bishops are getting around the recommendations of the ... commission so they don't have to face up to what is going to be a tsunami" of complaints, he said.
In May, Pope Francis passed a landmark new measure obliging anyone in the Church who knew about sex abuse to report it to their superiors.
A few months earlier, a French cardinal, Philippe Barbarin, received a six-month suspended jail sentence for failing to report sex abuse by a priest under his authority. His case is up for appeal later this month.
In August, the Vatican's former number three, Australian Cardinal George Pell, lost his appeal against his conviction for sexually assaulting two 13-year-old choirboys at a Melbourne cathedral in the 1990s.