The parents, from a small village outside Mandalay, forced their children to drink pesticide and attacked them with knives before trying in vain to take their own lives on Saturday.
The one-year-old baby girl died on the spot and the two boys, aged seven and nine, are now in hospital being treated for their injuries along with their parents.
Local lawmaker Sein Win said the impoverished couple had long had tuberculosis and became depressed after learning their infection was now resistant to treatment.
"They thought they were going to die because there was no treatment. They became crazed and tried to commit suicide with the whole family."
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Head of administration for Taungtha township, Wai Lin Tun, said the police were investigating the incident, but declined to give further details.
Myanmar has one of the highest rates of tuberculosis in the world, with more than 140,000 cases on record, according to the World Health Organisation.
An estimated 9,000 people are infected each year with a drug-resistant form of the disease.
But many of the rural poor struggle to keep up with the taxing treatment regimen, allowing the disease to develop resistance to treatment.
Tuberculosis, which is caused by a bacteria that infects the lungs and can be spread through the air, is one of the most deadly diseases in the world.
An estimated 1.8 million people died from tuberculosis in 2015, data from last year's WHO Global TB Report showed, making it among the top 10 causes of death worldwide.