A ministry statement offered its former employee "its sincerest congratulations".
"The new president can count on our determination to continue to act in favour of Georgia's sovereignty and territorial integrity in internationally recognised borders," it said.
War with Russia
Russia fought a war with Georgia in 2008 in support of two the two self-proclaimed republics of Abkhazia and South Ossetia, which have since been recognised by Moscow and five other UN member-states.
Official results with 99.9 percent of votes counted showed Zurabishvili, who was born in France to émigré parents, winning with 59.61 percent against her rival Grigol Vashadze's 40.46 percent.
But the 11-party opposition alliance has accused the ruling Georgian Dream party, which backed Zurabishvili, of intimidating voters, attacking its activists and fraud.
Saakashvili calls for protests
Former president Mikheil Saakashvili, who now lives in the Netherlands after being stripped of his Georgian citizenship, called for "mass peaceful protests".
He accused billionaire Bidzina Ivanishvili, who is widely seen as the country's de-facto ruler, of "stamp[in] out democracy".
Rights groups also claimed there had been widespread vote-buying and unprecedented fraud, including the alleged printing of fake ID cards.
Monitors from the Organisation for Security and Cooperation in Europe were to hold a press conference to announce their findings on Thursday.
Battle of the diplomats
Zurabishvili, 66, worked for the French diplomatic service and was appointed ambassador to Tbilisi in 2003.
She was later appointed foreign minister by Saakashvili, who fired her after a year after some MPs and senior diplomats accusing her of arrogance and impulsiveness.
Her dismissal prompted demonstrations of thousands.
Both she and Vashadze, also a former diplomat but for Georgia rather than France, favour Georgian membership of the European Union and Nato.
Although this is to be Georgia's last presidential election, it is seen as a test of Georgia's democratic credentials and could affect its candidacy to those organisations.
Turnout was 56.23 percent.
It will be followed by parliamentary elections next year.