"[Olmert] wished her success and the two agreed to meet. The prime minister said Livni will enjoy his full co-operation and Livni thanked the prime minister," the statement said.
Olmert will continue as the caretaker prime minister and Livni will now have 42 days to form a new government if she hopes to avert snap elections that polls say would bring the right-wing Likud party to power.
Exit polls in Wednesday's election from three different television stations showed Livni winning about 49 per cent of the vote, with a lead of at least 10 points over her main opponent Shaul Mofaz, the transport minister.
Polls opened at 10am (0700GMT). About 74,000 party members were eligible to vote and official results are expected at about 1am (1000GMT).
"People know they can make a change," Livni, a former Mossad spy, said after casting her ballot in Tel Aviv.
"The change begins here, at the polling booth," she said.
Livni, 50, who has been leading US-backed negotiations with the Palestinians, advocates an Israeli withdrawal from most of the occupied West Bank and east Jerusalem in order to reach a two-state solution with the Palestinians.
"I want to hold peace negotiations as long as the Palestinians want the same. But any agreement must provide security to Israel," she told supporters recently.
Al Jazeera's Jacky Rowland, in Tel Aviv, said that Livni was viewed as "a politician with clean hands, particularly since Israel has been reeling under the corruption scandals of Olmert.
"Israeli citizens want to wake up in the morning and read a newspaper that doesn't show their prime minister is involved in another financial scandal."
Rowland said that Israelis feel Livni brings "integrity, honesty and is someone without a whiff of scandal around her.
"She has been leading talks with the Palestinians for almost year now and wants to reach an agreement based on territorial concessions.
"She has said she hopes to get an outline agreement with the Palestinians by the end of the year."
Olmert agreed to relinquish Kadima leadership after being embroiled in scandals [AFP]
Israeli investigators are looking into allegations that Olmert accepted cash payments from an American businessman and double-claimed expenses for trips before he became prime minister.
The scandal forced Olmert to announce his resignation and sparked a contest for the party's leadership.
Kadima is trailing the Likud party of Benjamin Netanyahu, a former Israeli prime minister, in opinion polls.
Yehezkel Dror, a political analyst, said: "The problem is not what is happening at Kadima, but what will happen in the next elections ... we cannot predict how Netanyahu would behave if elected."
Kadima made the strongest showing in February 2006 elections just months after it was founded.
But corruption scandals and criticism of the government's handling of Israel's 34-day war in Lebanon in 2006 have undermined its rule.