He also laid out his plans for office, including waging war on corruption, rescuing the economy, bringing separatist territories back under control and mending the frosty relation with neighbouring Russia.
Official results from Sunday's presidential election have not been released but unofficial exit polls gave Saakashvili a massive lead and he said he had achieved a "sensational victory".
"This is a great responsibility," Saakashvili told reporters at a press conference at which he spoke in Georgian, Russian, English and French.
"We have got a very important mandate from the people to clean up Georgia, to make it peaceful and prosperous, to make Georgia efficient, investment-friendly, to consolidate power.
"We will do our best to promote these aims... You cannot do everything in one day. We will go step by step."
On foreign relations, Saakashvili said: "We are for European integration at the same time as close cooperation with the United States. We want to have just as close relations with Russia. That is an absolute priority for us."
He said he wanted to use the presidency to bring the Moscow-backed separatist region of Abkhazia back into the fold.
Abkhazia split off from Georgia in a war in the early 1990s along with the region of South Ossetia.
Former President Eduard
Shevardnadze was ousted last year
"It is neither in the interests of Georgia nor Abkhazia to leave the situation as it is," he said.
"We are determined to restart peace talks with Abkhazia," he continued. "We will never agree to it ceasing to be a part of Georgia."
"We hope for help from the United Nations and a constructive role from Russia. (Russian) President Vladimir Putin's statements give us hope that this will be the case. But the most important thing is to reach agreement with the Abkhaz people themselves."
On the domestic front, Saakashvili said he would soon set a date for a parliamentary election. He said he would share power with his allies Zurab Zhvania and Nino Burjanadze, who is caretaker president and is expected to resume her former post of parliament speaker. He said he would ask the parliament to create the new post of prime minister for Zhvania.
Asked about the future of Eduard Shevardnadze, the veteran leader Saakashvili ousted late last year in a popular uprising, Saakashvili said he "did not want to think about the past".
"We are concentrating on the future," he said.