The clashes broke out between the Myanmar military and Karen National Union in the town of Lay Kay Kaw near the Thai border after junta troops entered KNU territory and arrested several dissidents.
Hundreds of Myanmar villagers have fled to Thailand after junta troops clashed with an ethnic rebel group.
Fighting between Karen National Union (KNU) and the military broke out on Wednesday in the town of Lay Kay Kaw near the Thai border, the first such clashes in months.
"I heard small arms fire and heavy artillery," said one local from a village near Lay Kay Kaw.
"More people have arrived in our village and there are about 3,000 hiding here now."
On Thursday, photos by local media showed dozens of people lining up to cross a river that forms part of the border.
A Thai government official in Tak province who spoke on condition of anonymity told AFP 700 people had crossed the border, and would be given food and shelter.
Naw K'nyaw Paw, general secretary at the Karen Women's Organisation, told AFP that more than 1,000 had been allowed to cross after community leaders negotiated with Thai authorities.
The clashes broke out on Wednesday, a day after state media reported junta troops entered KNU territory and arrested several dissidents, including a former lawmaker from Aung San Suu Kyi's ousted government.
The group's fighters had since injured and killed junta troops, said Padoh Saw Thamain Tun of the KNU, without providing details.
AFP could not verify the claims and the junta did not respond to requests for comment.
The KNU has been a vocal opponent of the coup and provided shelter to dissidents working to oust the junta.
Its fighters have clashed sporadically with the Myanmar military along the Thai border.
In March, its fighters seized a military post and the army retaliated with air raids, the first in more than 20 years in Karen state.
Myanmar has more than 20 ethnic rebel groups, many of whom hold territories in the country's border regions.