Zida declared a rival claim by army chief Navere Honore Traore was "obsolete" in a televised speech.
The tussle for power comes after an extraordinary week of violent protest against Compaore's 27-year rule that saw parliament stormed and set ablaze.
Zida, who is leading a group of young army officers, said he had assumed "the responsibilities of head of the transition and of head of state" to ensure a "smooth democratic transition".
He told journalists that the former president, who was said to have fled the capital Ougadougou, was "in a safe place" and his "safety and well-being are assured".
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Earlier, Zida ordered the closure of the country's borders.
A French diplomatic source told AFP that Compaore was travelling south towards the town of Po near the border with Ghana.
The source said he was still in the country and had not asked for refuge in France, the former colonial power.
The uprising, which has drawn parallels with the Arab Spring, was sparked by plans to change the constitution to allow Compaore to stand once again for elections next year.
Army chief Traore stated on Friday that he was assuming power as head of state, a day after he ordered the dissolution of the government and a dusk-to-dawn curfew.
But many protesters are deeply opposed to him taking power, seeing him as a close ally of Compaore.
"We do not want General Traore in power. We need someone credible. Traore is Blaise Compaore's henchman," said Monou Tapsoaba, an activist with the opposition People's Movement for Progress.
One opposition leader, Benewende Sankara, described Traore's move as a "coup".