Mr Kabila has promised that the polls, which were supposed to have taken place two years ago, will be DR Congo's first orderly transfer of power since it gained independence from Belgium in 1960.
He will take office from President Joseph Kabila, who has ruled the resource-rich central African country since 2001.
Tshisekedi is the son of late opposition leader Etienne but is relatively unknown and untested.
It will come as a shock to many observers who believed authorities would ensure that the government candidate, Emmanuel Ramazani Shadary, would be the victor in the polls, the third since the end of a bloody civil war in 2002.
Leading opposition candidate Martin Fayulu has urged the electoral commission to announce the true results as quickly as possible and warned it not to "play with fire, it is very risky".
But Fayulu dismissed the results announced by the electoral commission as "a true electoral coup".
They would be deployed if needed to protect United States citizens and diplomatic facilities in DR Congo's capital, Kinshasa, he said.
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Kabila's choice of successor fuelled accusations that the long-time leader - concerned about possible retribution - would use Shadary to protect his interests after the vote. Election observers had reported numerous irregularities.
He blamed the delay on opposition parties' insistence that results be counted by hand and not transmitted electronically via voting machines, which Congo used for the first time.
"This is the coronation of a lifetime", the deputy secretary-general of Tshisekedi's party, Rubens Mikindo, said shortly after the announcement that his candidate had won, above the cheers at party headquarters.
Vote tallies compiled by the DRC's Catholic church found Fayulu clearly won the election, two diplomats told Reuters, raising the spectre of protests that many fear could lead to violence.
"These results have nothing to do with the truth at the ballot box", Fayulu is said to have told Radio France International.
Mr Nangaa, who has given no date for when provisional or final results would emerge, has said people need to be patient amid rising tensions over the delay. And in a last-minute decision, some 1 million of the country's 40 million voters were barred from participating, with the electoral commission blaming a deadly Ebola virus outbreak.
Anti-riot police with water cannon and armored vehicles are outside Congo's electoral commission ahead of the announcement of the first results of the presidential election. The Kinshasa lawmaker and businessman said his coalition would release its own figures if the announced results don't conform to the will of the Congolese people. He is known as the "people's soldier" for leading protests against President Kabila. He was less visible in campaigning than Fayulu and did not make himself available to reporters after the vote.