The CAPSAT unit has suspended key democratic institutions, including the constitutional court and the electoral commission, but has promised elections within two years
Lawmakers earlier voted to impeach Rajoelina, which he rejected as "null and void"
His whereabouts are unclear - there are reports he has fled the country on a French military plane and he said he was sheltering from a "safe place" after an attempt on his life
Madagascar has been rocked by weeks of protests - here's why young people have been demonstrating against Rajoelina's rule
Local media are reporting that the constitutional court in Madagascar has confirmed the leadership of Col Mickael Randrianirina.
The court also confirmed vacancies in the office of the president and the president of the senate.
This follows the national assembly - the lower house of parliament - voting to impeach President Rajoelina.
The elite military unit's announcement that it had suspended the activities of the Constitutional Court is significant.
The court was supposed to have approved the decision of the National Assembly - the lower house of parliament - to impeach Rajoelina, for it to take effect.
"We will form a government and reach consensus," Col Michael Randrianirina has told reporters in Madagascar after announcing that his elite unit, CAPSAT, had seized power.
Asked if they would hold elections, he replied: "Of course." This will happen within the next "18 months to two years", he said.
The CAPSAT chief said Gen Z Mada protesters would be part of the changes because "the movement was created in the streets so we have to respect their demands".
He said he and his fellow officers had no choice but to unseat the current regime.
"There is no president, there is no government... nothing here works."
The head of the elite unit that says it has seized power in Madagascar has announced that it would be suspending all constitutional institutions except the lower house of parliament, which earlier voted to impeach the president.
This includes the senate, the constitutional court and the electoral commission, the Reuters news agency reports.