"We have much to do and we will do it best by working together," Ms Gillard said.
"I am determined that 2013 will be the year when we together write the next great chapter in our nation's history."
Mr Abbott also called on Australians to approach 2013 with confidence, vowing the coalition offers plans that will deliver jobs and a strong and prosperous economy.
"Over the months ahead, my colleagues and I will be talking about how we can help all Australians to get ahead," he said.
"This is the positive choice."
The coalition was "not afraid" of having a go and would approach 2013 with a vision of a better Australia, he added.
His government would create two million new jobs, expand small business, clean up the environment, strengthen the borders and build modern infrastructure.
Lower taxes, more efficient government and higher wages would also be a benchmark of a coalition government, Mr Abbott said.
"Let us all approach the coming year with confidence, knowing that we are a people who can rise to any challenge and that there is a better way for our country," he said.
Ms Gillard said Australians had a lot to be thankful for as they start the new year, living in the "best country in the world" with regional security and a strong economy.
But the prospect of a nuclear Iran, the ongoing war in Afghanistan and missile tests in North Korea posed real risks.
Ms Gillard said Australia would use its new two-year term on the UN Security Council, which commences on January 1, to tackle these and other issues.
Alluding to the National Insurance Disability Scheme (NDIS) and the Gonski reforms, Ms Gillard said her government was ready to write the "next great chapter" in Australia's history.
"A chapter about jobs, opportunity, a better education for our children, a better deal for Australians with disability, and that we do so with a renewed Australian spirit," the prime minister said.
And if Ms Gillard needed a boost ringing in the new year, she could find some assurance in the last Newspoll of the year.
The poll, published in The Australian, found voters were the least pessimistic about their standard of living in the coming months than at any time since the early months of the Gillard government.
