We all have a responsibility to work for a resolution of this profound and extremely dangerous situation, a crisis that has potentially massive repercussions for the region and the world”
End QuoteBan Ki-moonUN secretary general
It also says the council will "consider further steps as appropriate", without specifying a time frame.
It is not binding and falls short of a formal resolution.
But the BBC's Nada Tawfik at the UN says diplomats hope it will intensify pressure on Mr Assad to work co-operatively with Mr Annan.
Diplomats said Western powers had agreed to soften the statement in order to gain the support of Russia and China, which had threatened to veto an earlier, tougher draft.
China and Russia have in the past blocked two resolutions by the council condemning Mr Assad's actions.
Mr Annan has spent the last few weeks meeting all sides in the conflict - putting forward proposals to try to bring about an immediate ceasefire by both sides, access for humanitarian aid and the beginning of political dialogue.
UN Secretary General Ban Ki-moon earlier warned of serious repercussions if the crisis in Syria is not resolved.
Speaking at a conference in Indonesia he warned the world could not afford to look away.
"We all have a responsibility to work for a resolution of this profound and extremely dangerous situation, a crisis that has potentially massive repercussions for the region and the world," he said.
In Syria itself, the British-based Syrian Observatory for Human Rights said the army had been active in the Khalidiya district of Homs. Activists have also reported shelling in Rastan, north of Homs, and at Qalat Mudiq, northwest of Hama.
Syria restricts access to foreign media which often makes it impossible to independently verify reports coming out of the country.
The UN says more than 8,000 people have been killed in the year-long uprising, while tens of thousands of people have fled their homes.
On Tuesday, Russia - a key ally of Damascus - warned Syria's leadership it was making "a lot of mistakes", signalling Moscow may be hardening its stance on Damascus.
The same day US campaign group Human Rights Watch (HRW) accused elements of Syria's armed opposition of carrying out serious human rights abuses, including kidnapping, torture and execution.
The opposition Syrian National Council said in a statement that it "deplores the reported incidents of human rights violations by armed opposition groups in Syria" and it is working to ensure "abuse does not happen in the fight for freedom".
HRW has frequently accused Syria's government of abuses during the conflict.
The Syrian government has been trying to suppress an uprising inspired by events in Tunisia, Egypt and Libya. The UN says thousands have been killed in the crackdown, and that many more have been detained and displaced. The Syrian government says hundreds of security forces personnel have also died combating "armed terrorist gangs".
The family of President Bashar al-Assad has been in power since his father, Hafez, took over in a coup in 1970. The country underwent some liberalisation after Bashar became president in 2000, but the pace of change soon slowed, if not reversed. Critics are imprisoned, domestic media are tightly controlled, and economic policies often benefit the elite. The country's human rights record is among the worst in the world.
Syria is a country of 21 million people with a Sunni Muslim majority (74%) and significant minorities of Alawites - the Shia heterodox sect to which Mr Assad belongs - and Christians. Mr Assad promotes a secular identity for the country, but he has concentrated power in the hands of family and other Alawites. Protests have generally been biggest in Sunni-dominated areas.
Under the sanctions imposed by the Arab League, US and EU, Syria's two most vital sectors, tourism and oil, have ground to a halt in recent months. The IMF says Syria's economy contracted by 2% in 2011, while the value of the Syrian pound has crashed. Unemployment is high, electricity cuts trouble Damascus, and critical products like heating oil and staples like milk powder are becoming scarce.
Pro-democracy protests erupted in March 2011 after the arrest and torture of a group of teenagers who had painted revolutionary slogans on walls at their school in the southern city of Deraa. Security forces opened fire during a march against the arrests, killing four. The next day, the authorities shot at mourners at the victims' funerals, killing another person. People began demanding the overthrow of Mr Assad.
The government has tried to deal with the situation with a combination of minor concessions and force. President Assad ended the 48-year-long state of emergency and introduced a new constitution offering multi-party elections. But at the same time, the authorities have continued to use violence against unarmed protesters, and some cities, like Homs, have suffered weeks of intense bombardment.
The opposition is deeply divided. Several groups formed a coalition, the Syrian National Council (SNC), but it is dominated by the Sunni community and exiled dissidents. The SNC disagrees with the National Co-ordination Committee (NCC) on the questions of talks with the government and foreign intervention, and has found it difficult to work with the Free Syrian Army - army defectors seeking to topple Mr Assad by force.
International pressure on the Syrian government has been intensifying. It has been suspended from the Arab League, while the EU and the US have imposed sanctions. However, there has been no agreement on a UN Security Council resolution calling for an end to violence. Although military intervention has been ruled out by Western nations, there are increasing calls to arm the opposition.
Correspondents say a peaceful solution seems unlikely. Syria's leadership seems intent on crushing resistance and most of the opposition will only accept an end to the regime. Some believe the expected collapse of Syria's currency and an inability to pay salaries may be the leadership's downfall. There are fears, though, that the resulting chaos would be long-lasting and create a wider conflict.
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