Mr Turnbull, who was communications minister under Mr Abbott, is the fourth prime minister since 2013.
After taking the oath of office, Mr Turnbull attended his first Question Time in parliament as prime minister.
Mr Abbott on Tuesday said his removal was "tough" but promised not to undermine the new government.
In his first public comments since his removal by the Liberal party late on Monday, Mr Abbott said it had been "a tough day, but when you join the game, you accept the rules".
He said he was proud of what his government had achieved, while also taking swipes at party members who had leaked to the media and carried out "a sour, bitter character assassination".
He did not say if he would remain on the backbench or eventually resign from politics.
Mr Abbott's removal followed weeks of tumbling polls ratings and speculation about a challenge to his leadership.
In a move led by Mr Turnbull and his deputy and Foreign Minister Julie Bishop, Mr Abbott was voted out by the Liberal party by 54 votes to 44.
In his first Question Time session in parliament, Mr Turnbull paid tribute to Mr Abbott, and said these are "the most exciting times to be an Australian".
"The future is one of great opportunities and that requires confidence and leadership and it will be lost if we embrace the politics of fear and scaremongering."
He said policies would "change in the light of changed conditions", but did not signal any immediate amendments to contentious issues including same sex marriage and climate policy.
Opposition Labor leader Bill Shorten said Mr Abbott had been "a formidable fighter" in office, and praised his "generous and personal" nature.
Mr Turnbull is not expected to announce a new cabinet line-up until the end of the week.
But following his fierce criticism of Mr Abbott's economic leadership, there has been speculation that Joe Hockey might lose the Treasury portfolio.
Ms Bishop, who has retained her role as deputy, said that Mr Abbott had "done a fantastic job in winning the 2013 election" but that "a number of people felt that he hadn't met their expectations".
After surviving another leadership challenge in February, Mr Abbott had asked for six months to turn the Liberal Party's electoral hopes around, she said.
"Now seven months later the majority have decided they wanted a change of leader and that he had lost their confidence."
But she said Mr Abbot was "obviously very hurt".
Mr Turnbull said on Monday night the government would serve a full term, meaning a general election is likely in mid-2016.
The first real test of how the public has taken the news will be on Saturday in a by-election for the seat of Canning in Western Australia.
The seat is held by the Liberal Party. Opinion polling done before Monday night's ballot suggested the Liberals would retain the seat but at a reduced majority.
Malcolm Turnbull has been sworn in as Australia's new prime minister, after Tony Abbott was ousted by his party in a leadership challenge.