It called on the Damascus government to cooperate with UN human rights officials and said "those responsible for the violence should be held accountable."
The council also demanded that Assad, who was not mentioned by name, carry out promised reforms.
Arab council member Lebanon, however, swiftly disavowed the statement saying it would not help end the crisis in its neighbor, where activists say more than 1,600 people have been killed since protests started in mid-March.
But Britain, France, Germany and Portugal, which led the campaign for action by the 15-nation council, insisted the statement sent a strong warning.
There is "a clear, unambiguous and united message to the Syrian regime: barbarous acts must cease in Syria, the country must find its way onto a path of stability," Britain's UN ambassador Mark Lyall Grant said.
The international community is "unanimously condemning the murderous actions of the Syrian authorities against their own people," added France's deputy ambassador Martin Briens.
UN Secretary General Ban Ki-moon welcomed the statement calling recent events in Syria "brutally shocking."
The Security Council and Ban demanded an immediate end to the violence in Syria, as government tanks Wednesday entered the flashpoint city of Hama, where scores of people have been killed in recent days.
The European powers, with US backing, first proposed a UN resolution on Syria more than two months ago.
Until a weekend surge in the violence, permanent members China and Russia had threatened to veto any formal resolution. India, Brazil and South Africa had also opposed council action, saying it could lead to a Libya-style military intervention by the West.
Russia, China and others maintained their opposition to a formal resolution. The statement that was adopted in the name of the council president still carries decision-making weight.
Normally the document is adopted unanimously. Lebanon did not block the adoption, but disavowed the document after.
"Whatever affects Lebanon affects Syria, whatever affects Syria will also affect Lebanon," Lebanon's deputy ambassador Caroline Ziade told the meeting.
"Since Lebanon considers that the statement being discussed today does not help address the current situation in Syria, Lebanon therefore disassociates itself from this presidential statement," she added.
Diplomats said Lebanon was a special case because of its sensitive ties to Syria. China and the United States have in the past disassociated themselves from council statements.
US ambassador Susan Rice said the "strong content" was more important than whether the council passed a resolution or statement.
"What was most important from a US point of view was a clear and unequivocal condemnation of the Syrian authorities for the abhorrent and crazy violence they perpetrated against their own people," she told reporters.
Rice added the statement would come as a shock to the Syrian government.
"I think the Assad regime has been counting on the fact that the Security Council would be unable to speak, and that they would not be condemned, and that they would have protectors and defenders that would make it impossible for that condemnation to emerge," she said.
"Surely they must be quite surprised and disappointed by the outcome."