Abbott, head of the Liberal-National coalition, claimed victory, saying, "I can inform you that the government of Australia has changed."
With two-thirds of the votes counted, the Australian Broadcasting Corporation said the coalition had secured 85 seats in parliament and was on course to win 91. Labor had secured 54, with a predicted final tally of 55.
The dramatic change of fortune for Rudd came months after he ousted Julia Gillard as Prime Minister and Labor leader.
Earlier, outgoing Prime Minister Rudd conceded defeat and announced that he would not re-contest the leadership of the Labor Party, saying it was "time for renewal".
In his address after victory, Abbott repeated his election pledges, saying the carbon tax would go, the boats would be stopped and the budget would be on track for a "believable surplus".
"From today, I declare that Australia is under new management and that Australia is once more open for business," he said.
The new Prime Minister elect said that in a "week or so" Governor-General Quentin Bryce would swear in the new government.
"Today, the people of Australia have declared that the right to govern this country does not belong to Mr Rudd or to me or to his party or to ours; but it belongs to you, the people of Australia," Abbott said to a swarm of cheering crowd.
"I gave it my all but it was not enough for us to win," Rudd told a crowd of cheering Labor Party supporters in Brisbane.
Rudd said he was proud he had helped preserve Labor as a "viable fighting force" for the future.
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