The agency reported the deaths in a bulletin late on Sunday while state radio said some died when their house collapsed in the town of Ambalavao, about 460km (286 miles) south of the capital Antananarivo.
It was the second major storm to hit the poor Indian Ocean island nation in two weeks, AlJazeera reported.
The cyclone made landfall in Mananjary, with winds of 165 kilometres (103 miles) per hour, uprooting trees, destroying buildings and forcing residents to weigh down flimsy corrugated iron roofs along its path.
The Meteo-France weather service had earlier predicted Batsirai would pose a “very serious threat” to Madagascar, after passing Mauritius and drenching the French island of La Reunion with torrential rain.
Some 10,000 people on La Reunion were still without electricity on Sunday, three days after the tropical cyclone passed through the island, injuring 12 people on its path.
Tropical Storm Ana had affected at least 131,000 people across Madagascar in late January, with nearly 60 people killed, mostly in the capital Antananarivo.
Ana had also hit Malawi, Mozambique and Zimbabwe, causing dozens of deaths.
Meteo Madagascar, the national meteorological office, said that Batsirai had weakened as it crossed the country and that the average wind speed had halved.
ZZ/PR