Ghannouchi is now the interim president.
He cited chapter 56 of the Tunisian constitution as the article by which he was assuming power.
Maltese air traffic controllers have told Al Jazeera that Ben Ali is bound for Paris, though the Maltese government has denied any knowledge of Ben Ali's plane having stopped in Malta after having left Tunis.
In his televised address, Ghannouchi vowed to respect the constitution and restore stability, and called on citizens to "maintain patriotic spirit ... in order to brave through these difficult moments".
He also vowed to carry out inflation and unemployment redressal policies "exactly" as they had recently been announced by Ben Ali.
Ayesha Sabavala, a Tunisia analyst with the Economist Intelligence Unit in London, told Al Jazeera that with President Ben Ali out of the country, there are "only ... a few people ... capable of [running the country] within the RCD [the ruling Rassemblement Constitutionel Démocratique party], and Ghannouchi is an ideal candidate".
Abdel Karim Kebiri, a former senior adviser to the International Labour Organisation, told Al Jazeera that "the people will be happy" with Ben Ali's departure.
Al Jazeera's Hashem Ahelbarra said the days events were a "political earthquake" and "something unheard of".
He said that it was also as yet unclear what role the army was playing in matters, as it was reportedly moving towards the capital to "take charge of the situation from the security forces, which are seen by Tunisians as a main problem for its long record of human rights abuses".
Friday's developments come following weeks of violent clashes across the country over unemployment and rising food prices.
Matters came to a head in the capital, Tunis, on Friday, as police tear-gassed protesters gathered outside the interior ministry building. Witnesses said police used batons to disperse the crowd, but the protesters insisted they would not leave until Ben Ali steps down.
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"Simply bringing in an interim president, and especially one who has been close to Ben Ali ... is not going to be enough," she said.
"Logically, there is bound to be a lot of distrust, because Ghannouchi is part of the very close inner circle ... of Ben Ali. Past promises that have been made [by that government] have not been kept."
State media earlier reported that Ben Ali had imposed a state of emergency in the country and promised fresh legislative elections within six months in an attempt to quell the wave of dissent sweeping across the country.
There were also reports that the airport in Tunis had been surrounded by troops and the country's airspace has been closed. Air France, the main international airline into and out of Tunisia announced that it had ceased flights to Tunisia following that announcement.
State TV also reported that gatherings of more than three people had been banned.
Ben Ali had been in power for the last 23 years. On Thursday, he vowed not to seek re-election and reduce food prices in a bid to placate protesters.
But the pledges seemed to have little effect as fresh street protests erupted on Friday.
The unrest in the country began on December 17, after a 26-year-old unemployed graduate set himself on fire in an attempt to commit suicide. Mohammed Bousazizi's act of desperation set off the public's growing frustration with rising inflation and unemployment, and prompted a wave of protests across the country.