The resolution welcomes the deployment of a force to fight armed Islamists, people traffickers and drug smugglers that the G5 group of African countries - Burkina Faso, Chad, Mali, Mauritania and Niger - agreed to form in March.
But it drops a provision that invoked chapter 7 of the UN charter, which authorises the use of force and a call for a report from UN Secretary-General Antonio Guterres outlining the logistical and financial needs.
That was because of objections by the US, the leading financial contributor to UN peacekeeping operations, because it wants to scale back funding.
The resolution instead requests that Guterres report on "challenges encountered and possible measures for further consideration" in the coming months, a formulation that still leaves the door open to funding in the future.
In the meantime, the European Union has agreed to give 50 million euros to the force and a pledging conference will be held to raise further finance.
But responsibility for providing the troops with adequate resources lies with the G5 countries, according to the resolution.
France welcomes vote
French Foreign Minister Jean-Yves Le Drian said the council had expressed "its full, clear and total support" for the African force and called for international assistance to the troops.
French Ambassador Francois Delattre said the unanimous vote was proof of "maximum political support" for the force.
It will have its headquarters in Mali but will be under a separate command from the UN peacekeeping force Minusma and work in coordination with France's 4,000-strong military presence in the region, known as Barkhane.