In a unanimous statement, the council called for an immediate halt to attacks against civilians and stressed the importance of upholding human rights.
The appeal from world powers came a week after Russia and China had blocked a similar draft statement on the crisis in Sudan.
More than 100 people were killed in a crackdown on protesters on June 3, according to a committee of doctors, but officials have said the death toll is much lower, AFP reported.
Sudan has been led by a military council since it toppled president Omar al-Bashir on April 11 after months of nationwide protests against his autocratic rule.
As protests demanding an end to military rule continued, Sudan's security forces violently broke up a sit-in outside the military headquarters in Khartoum on June 3.
Following a mediation led by Ethiopia, protest leaders agreed to return to talks with Sudan's generals.
The council called on all sides "to continue working together towards a consensual solution to the current crisis" and voiced support for African-led diplomatic efforts.
UN diplomats said the statement drafted by Britain and Germany ran into some resistance from China and Russia, which did not want to issue a condemnation, but the strong language was nevertheless agreed.
A senior US diplomat, Tibor Nagy, is to arrive in Khartoum this week for talks on the crisis.
Negotiations between the military rulers and protest leaders had broken down over disagreements on whether a planned transitional body would be headed by a civilian or a military figure.
The United Nations is strongly backing the African Union in its bid to put Sudan on a path toward civilian rule.
The Security Council will discuss Sudan on Friday during a meeting focused on the joint UN-AU peacekeeping mission in Darfur, known as UNAMID.