Monday's shooting came as support for an international military response was mounting if it is confirmed that Assad's troops used chemical weapons. France, Britain, Israel and some U.S. congressmen have said such a response against the Syrian regime should be an option.
Russia, meanwhile, said Western nations calling for military action have no proof the Syrian government was behind any chemical attacks.
News of the sniper incident came only a few hours after an Associated Press photographer saw the team members wearing body armor leave their hotel in Damascus in seven SUVs and head to the site of the alleged attack.
The photographer said U.N. disarmament chief Angela Kane saw them off as they left but did not go with them.
The Syrian government said its forces provided security for the team until they reached a position controlled by the rebels, where the government claimed the sniper attack occurred.
Martin Nesirky, spokesman for U.N. Secretary-General Ban Ki-moon, said one of the U.N. vehicles was "deliberately shot at multiple times" Monday in the buffer zone area between rebel- and government-controlled territory, adding that the team was safe.
Nesirky said the car was "no longer serviceable" after the shooting, forcing the team to return to a government checkpoint to replace the vehicle.
In a statement broadcast on Syrian TV, the government said it "holds the terrorist gangs responsible for the safety of the United Nations team." The Syrian regime routinely refers to rebels fighting to topple Assad as terrorists.
Wassim al-Ahmad, a member of the Moadamiyeh local council, and the main Syrian opposition group in exile, the Syrian National Coalition, said members of a pro-government militia known as the Popular Committees fired at the U.N. team to prevent them from going in.
The rebel coalition said the sniper shots occurred near the final checkpoint between rebel and regime-controlled areas, calling it an attempt by the regime "to intimidate the U.N. team and prevent it from discovering the truth about Assad's chemical weapons attack against civilians."
U.N. chief Ban said in a statement Monday that he had "instructed Angela Kane to register a strong complaint to the Syrian Government and authorities of opposition forces."
Al-Ahmad said five U.N. investigators eventually arrived at a makeshift hospital in the suburb, where doctors and about 100 people still suffering symptoms from the alleged chemical attack were brought in to meet with the U.N. team.
He said the U.N. experts took blood, hair and tissue samples with them, after which heavy shelling of Moadamiyeh resumed.
"They are late, they came six days late," al-Ahmad told AP via Skype from Moadamiyeh, referring to the time it took the U.N. team to arrive. "All the people have already been buried."
The eastern suburbs of Damascus have also witnessed a wide army offensive over the last week, but have been relatively quiet since Sunday night, said the U.K.-based Syrian Observatory for Human Rights, which tracks the fighting in Syria.
Mohammed Abdullah, an activist in the eastern suburb of Saqba, said the U.N. is expected to visit that rebel-held area on Monday and will be under the protection of the Islam Brigade, which has thousands of fighters in the area.
Syrian activists and opposition leaders have said that between 322 and 1,300 people were killed in the alleged chemical attack in the capital's western suburbs.
Syria said Sunday that a U.N. team would be allowed to investigate the site, but a senior White House official dismissed the deal as "too late to be credible."
U.S. Defense Secretary Chuck Hagel, speaking to reporters Monday after meeting with his Indonesian counterpart, said Monday the administration of U.S. President Barack Obama was studying intelligence on Syria's purported use of chemical weapons and "will get the facts" before acting.
Hagel said Obama "is considering all different options" and that "if there is any action taken, it will be in concert with the international community and within the framework of a legal justification."
Assad told Russia's Izvestia daily that accusations that his troops used chemicals were "politically motivated."
"This is nonsense," Assad was quoted as saying in the interview published Monday. "First they level the accusations, and only then they start collecting evidence."
Assad said that attacking such an area with chemical weapons would not make sense for the government as there was no clear front line between regime and rebel forces.
"How can the government use chemical weapons, or any other weapons of mass destruction, in an area where its troops are situated?" he asked.
The U.N. team's conclusions could have a dramatic impact on the trajectory of the country's civil war.
Speaking to reporters in the South Korean capital of Seoul, the U.N. secretary-general said that "any use of chemical weapons by anyone under any circumstances is a serious violation of international law and an outrageous crime. We cannot allow impunity in what appears to be a grave crime against humanity."
French Foreign Minister Laurent Fabius said no decision had been made on a military intervention but that any response would be "proportionate."
"It will be negotiated in coming days," Fabius told Europe 1 radio on Monday. He said that the lack of a U.N. blessing was problematic, but that all options remain on the table.
"The only option that I can't imagine would be to do nothing," Fabius said.
Britain's Foreign Secretary William Hague said diplomatic pressure has not worked on Syria's government, adding that a response was still possible without complete unity in the U.N. Security Council.
Also Monday, the German government suggested for the first time that it would support an international military response it the attack were confirmed.
Chancellor Angela Merkel's spokesman Steffen Seibert said Monday that if U.N. inspectors learn chemical weapons were used, "it must be punished."
Turkish Foreign Minister Ahmet Davutoglu said Turkey would take part in an international coalition against Assad if the U.N. failed to come up with sanctions to punish Syria for the alleged use of chemical weapons. Turkey has been one of Assad's harshest critics.
Russian Foreign Minister Sergei lavrov, however, said the countries calling for military action have assumed the role of "both investigators and the U.N. Security Council" in probing the incident.
Lavrov likened the situation in Syria to the run-up to the 2003 military operation in Iraq. He said "the use of force without a sanction of the U.N. Security Council is a crude violation of the international law."
Assad said in the interview that a military campaign against his country will not succeed.
"They can start a war but they will not know where it will spread or how it will end," Assad said. "Superpowers can launch wars but they cannot win them."
Asked what the US. would face in any intervention, Assad answered: "What it suffered in all its wars from Vietnam until now. Failure."