On Thursday Tshisekedi was named by election officials as the provisional victor of the country's long-delayed, chaotic and controversial presidential poll, in a surprise announcement that appeared to contradict both pre-election surveys and the findings of independent monitors.
Tshisekedi, who received more than 7 million votes, had not been widely considered the leading candidate.
Tshisekedi took 38.57 percent of the vote, while his closest challenger, Martin Fayulu, garnered 34.8 percent, according to the electoral board.
He joined forces with Vital Kamerhe, the head of another major opposition party who came third in the 2011 election, in an effort to boost his popularity in eastern Congo.
The constitutional court has 14 days to validate them.
Already delayed by two years, the announcement of results was postponed by a further week to allow more time to overcome logistical challenges in a country of 80 million inhabitants spread over an area the size of western Europe with nearly no paved roads.
If confirmed, Mr Tshisekedi will be the first opposition challenger to win since the DR Congo gained independence.
Kabila, 47, has ruled since the 2001 assassination of his father, Laurent Kabila, who overthrew long-serving dictator Mobutu Sese Seko in 1997.
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Prior to Wednesday's electoral announcement, fears of violence had been escalating as dragged out vote-counting continued after the December 30 elections, with the official electoral body CENI blaming the delay on manual vote counting despite Sunday being the official day for the results to be announced. The electoral commission said this was due to an Ebola outbreak and violence there, but it means about a million people have been disenfranchised because the new president is due to be sworn in this month.
Mr Fayulu, a former oil tycoon, dismissed the results as "a true electoral coup", telling Radio France Internationale they had "nothing to do with the truth of the ballot box".
And this is not the first sign of discontent between Fayula and Tshisekedi, which began even before the presidential elections.
Congo's powerful Catholic Church has upped the stakes by saying it found a clear victor from data compiled by its 40,000 observers deployed to all polling stations. The United States threatened sanctions against officials who rigged the vote.
How will Congolese citizens react?
NPR's Ofeibea Quist-Arcton reports from Kinshasa that the results could lead to protests and allegations of rigging.
On Wednesday afternoon, hours before results were announced, some Tshisekedi supporters began to celebrate at his Union for Democracy and Social Progress party headquarters, with calendars already printed saying "Felix Tshisekedi president".