The violence occurred as Christian militia tried to block the evacuation of Muslims and disrupted a visit by a top United Nations aid official, witnesses said.
The fighting was the latest episode in a cycle of religious violence that has displaced 1 million people from the former French colony and killed at least 2,000. About 6,000 African and 1,600 French peacekeepers have failed to halt the conflict.
Paris has warned the crisis risks degenerating into genocide.
The latest clashes began after Chadian troops tried to escort a convoy of Muslims out of the city, said Songokoua Yetinzapa, a Bangui resident living in a vast camp for displaced civilians near the airport. Their departure was blocked by the militia, known as the "anti-balaka".
"I heard several people were killed but I only saw one dead body: a Muslim who was killed by the anti-balaka," he said by telephone as automatic gunfire and an explosion rang out.
Reuters journalists were chased from the scene by machete-wielding youths.
A spokesman for the anti-balaka said their fighters came out to defend the local population near airport after they were targeted by Chadian troops who were escorting the convoy.
"When they drove in from Chad yesterday, the Chadians attacked civilians in Damara. Someone from the town called a radio programme to report what the Chadians did. This angered people in Bangui who set up road blocks to prevent them from leaving," Sebastien Wenezoui told Reuters by telephone.
Wenezoui said African MISCA peacekeeping troops and French soldiers removed the roadblocks and dispersed the crowds. Some anti-balaka fighters responded with gunfire.
Chadian troops then "passed forcefully" through the barricades, a foreign military source said, asking not to be named. French army spokesman Gilles Jaron said that there was sporadic gunfire in the direction of French troops but gave no further details.
A UN official confirmed African peacekeepers had been dispatched to reinforce troops in the area near the airport.
Another UN official said the fighting had prevented Valerie Amos, U.N. Under-Secretary-General for Humanitarian Affairs, from travelling to the north of the country, where violence between Muslims and Christians has also scattered tens of thousands of civilians and left villages abandoned.
Predominately Muslim rebels, known as Seleka, toppled the president of the Central African Republic, Francois Bozize, last March. Killings and abuses carried out while they were in power led to the creation of a mainly Christian self-defence militia known as "anti-balaka".
UN officials estimate around 1 million people have been displaced by the violence and at least 2,000 people killed since the start of the crisis. But they say the actual death toll will rise much higher as mass graves are unearthed and killings continue.
Violence has escalated since the Seleka leader Michel Djotodia quit power in January under international pressure.
Seleka's retreat has led to Muslims fleeing the south and warnings from a top U.N. official of "ethnic-religious cleansing".