Although the official result has not yet been declared, main rival Mar Roxas admitted defeat after polls gave Mr Duterte an unassailable lead.
Mr Duterte said he accepted the mandate with "extreme humility".
The 71-year-old stirred controversy during campaigning with his incendiary comments.
He has credited his success to his tough stance on law and order.
His record as the crime-crushing mayor of the southern town of Davao, once notorious for its lawlessness, earned him the moniker The Punisher and resonated with voters.
As the extent of his lead became clear, Mr Duterte told AFP news agency: "It's with humility, extreme humility, that I accept this, the mandate of the people."
Mr Roxas accepted his rival's win. "There are many tears in the room. Let me tell you this is not a time for tears. For our country, we have had a peaceful, successful transfer of power."
Another key rival, Senator Grace Poe, was the first to concede defeat, promising to "co-operate with the healing process" after a turbulent campaign.
In the election to be vice-president, Leni Robredo, a social activist, is currently slightly ahead of Ferdinand "Bongbong" Marcos, the son of a former dictator Ferdinand Marcos.
The statistics are stark, and bleak. Experts say anywhere between 40% and 80% of Philippine legislators are connected to political dynasties with vested interests. A handful of the families control almost all of the country's wealth.
And perhaps the bleakest statistic of all: 25% of the population lives under the poverty line - a figure that has not changed much in two decades.
So it shouldn't come as a surprise then that Filipinos have chosen Mr Duterte's crime-busting, tough-talking, action-oriented ways over the current administration's candidate Manuel Roxas.
Mr Aquino is standing down as the constitution limits presidents to one six-year term. As Mr Duterte rose in opinion polls ahead of voting, Mr Aquino had tried to unite other candidates against him, warning his election could mean a return to dictatorship for the Philippines.
Mr Duterte has made many controversial statements during his campaign, saying that he would massacre criminals and drug-users, joking about the rape of an Australian missionary and saying he would dissolve congress if it didn't agree with him.
He has also given forthright opinions on the Philippines' territorial dispute with China in the South China Sea, saying he would sail to disputed islands and plant the Philippine flag there.
The Philippines has taken one of its claims to a court of arbitration at the Hague. On Monday, he said he would seek multilateral talks over the issue, to include the US and Japan.
Source:BBC NEWS
