All countries around Syria are worried about fallout from the uprising, Annan told reporters after briefing a closed session of the UN Security Council on the crisis.
"We tend to focus on Syria, but any miscalculation that leads to major escalation will have impact in the region which would be extremely difficult to manage," he told reporters in Geneva, according to an official transcript.
"The region is extremely concerned about developments in Syria. Their concern goes beyond Syria itself because the crisis can have a serious impact for the whole region if it is not handled effectively."
Turkey Friday strongly urged thousands of nationals to leave Syria because of the "serious security risks" and said it was considering withdrawing its ambassador from Damascus.
Annan told the Security Council he has had a "disappointing" response from Assad so far to his proposals, diplomats at the meeting said.
He appealed to the badly divided council to get behind his mission in a bid to sway Assad.
"The stronger and more unified your message, the better chance we have of shifting the dynamics of the conflict," Annan was quoted as telling the 15-nation council.
The United Nations says some 8,000 people have died since opposition protests flared last March. Activists have given a toll of 9,100.
But the Security Council has been unable to pass a resolution condemning the violence. Russia and China have twice used their powers as permanent members to block resolutions which they said were unbalanced.
Following Annan's intervention, talks are expected to start again on a draft text drawn up by the US. Negotiations halted last week because of new objections from Russia and China.
Moscow says the onus the new draft places on the government to halt violence first is unreasonable in a conflict that is now as much armed insurgency as peaceful protest movement.
Syria's foreign ministry said the government would pursue its crackdown on "terrorism" while cooperating with Annan.
The government is "determined to protect its citizens by disarming the terrorists and continues to search for a peaceful solution to the crisis by cooperating with special envoy Kofi Annan," it said in a letter to the United Nations, reported by state news agency SANA.
Russian Foreign Minister Sergei Lavrov said Moscow was using its contacts in Syria to urge it to fully cooperate with Annan.
"Other Security Council members should also do their job and demand the opposition not provoke an escalation of the situation," he said.
Thousands of anti-government protesters called Friday for foreign military intervention to bring down Assad. The Syrian Observatory said at least 15 people were killed around the country.
"The people want military intervention, the Free Syrian Army to be armed, and the fall of the regime," several thousand demonstrators chanted in Aleppo in northern Syria, an activist at the scene told AFP by telephone.
Thousands of others took part in protests in the flashpoint provinces of Homs and across Daraa as well as several districts of Damascus and the region, said Rami Abdel Rahman of the Syrian Observatory for Human Rights.
Huge rallies in support of Assad were held in Damascus and other major cities on Thursday to mark the first anniversary of the uprising. Numbers have fallen at anti-regime demonstrations as security forces seize protest centers.
Annan's experts are expected in Damascus on Sunday. The United Nations and Organization of Islamic Cooperation are also to send experts this weekend on a Syrian government-led humanitarian mission to Homs, Daraa and Hama.
The United Nations estimates more than 30,000 Syrians have fled to neighboring states and another 200,000 are displaced in Syria.