This could be Congo's first peaceful, democratic transfer of power since independence in 1960.
Tshisekedi had won with 38.57 percent of more than 18 million ballots cast, Corneille Nangaa, head of the election commission said at about 3am (02:00 GMT) in a news conference that appeared timed to avoid any immediate reaction in the streets.
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.
The Tshisekedi's victory is deeply controversial as pre-election polls had given outspoken opposition frontrunner Martin Fayulu, a respected former business executive, a healthy lead.
Ramaphosa and Lungu urged the DRC's electoral commission to "speedily finalise the vote tabulation and release the election results in order to maintain the credibility of elections", according to a statement from South Africa Department of International Relations and Cooperation seen by RFI.
"The CENI plans to devote 24 to 48 hours to this task, after which time the CENI will schedule the announcement of the provisional results", he said. "Where did the extra seven million votes come from [for Tshisekedi's victory]?" I ask the Cenco, the Church of Christ in Congo, the Symocel and all those who observed the elections to tell us the truth, to publish the results.
In order to do so they must appeal to Congo's constitutional court, which has 10 days to hear and rule on any challenges. Kabila has promised a peaceful transition of power.
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On November 11, Tshisekedi joined six other opposition leaders to rally behind a single unity candidate, Martin Fayulu, to take on Shadary.
"There's no spirit of revenge", said Tshisekedi. The delay in announcing results has led some Congolese to suspect possible manipulation in favor of the ruling party.
After several days of intense speculation on the outcome of the DRC's presidential elections, the country's election board, CENI, proclaimed Felix Tshisekedi the victor.
Attention now turns to Congo's powerful Catholic church, which has said its 40,000 election observers at all polling stations found a "clear winner" but was barred by electoral regulations from saying more.
The vast, mineral-rich Central African country is choosing a successor to departing President Joseph Kabila, in power since 2001.
Congolese riot police take position around the electoral commission building at night in Kinshasa, Congo, Tuesday Jan. 8, 2019.
More than one million people were kept from voting on election day because of an Ebola outbreak and militia violence in opposition strongholds, mainly in DRC's east.
"You know that this proclamation is the fruit of results trafficked, invented and fabricated", Fayulu told reporters after the results were announced.
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