About 32 people, including 27 civilians, three policemen, and two local paramilitary troopers were injured in the attack.
"Nineteen [people] were killed in fighting", the military source said, adding that two-dozen had been injured.
China's embassy in Myanmar said in a statement that the conflict had sent stray bullets into China, along with an unidentified number of people seeking refuge.
The conflict, including in the northern-most state of Kachin, has escalated over the past month as the worldwide focus remains on the Rohingya crisis in the west of the country. "This is a terrorist attack".
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Burma is often called Myanmar, a name that military authorities adopted in 1989.
The TNLA claimed responsibility for the attack in a statement posted on Facebook saying they were a response to recent Myanmar army offensives against another armed group, the Kachin Independence Army, or KIA.
TNLA spokesman Mai Aik Kyaw told Agence France-Presse that they attacked joint military and militia posts in the Shan state town of Muse and on a road to Lashio. It says that the insurgents are based in Shan and the neighboring state of Kachin, and are demanding greater autonomy.
Muse in Shan state is situated at a crossing between Myanmar and China's Yunnan province, a major source of trade between the two countries.
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Aye Myint, a police officer in Muse, said the death toll was 19, and that Ta'ang National Liberation Army fighters used small arms as well as mortars.
He said the military would pursue the armed group.
China on Sunday condemned fighting in Myanmar between Myanmar government troops and ethnic militants near the Chinese border, which had caused people to flee into Chinese territory.
The country's de facto leader, Aung San Suu Kyi, made ending Myanmar's multiple civil wars, which have lasted for 70 years, a priority when her National League for Democracy won landslide elections in 2015, ending five decades of military rule.
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