Provisional results put him ahead of another opposition candidate Martin Fayulu, and the ruling coalition's Emmanuel Chaudhary.
If confirmed, Tshisekedi will be the first opposition challenger to win since the DR Congo gained independence.
Current President Joseph Kabila is stepping down after 18 years in office.
He had promised DR Congo's first orderly transfer of power since the country's independence from Belgium in 1960.
The election outcome was initially expected to be announced on Sunday. The interim result can still be challenged.
In the early hours of Thursday the head of DR Congo's National Electoral Commission (Ceni), Corneille Nangaa, said Mr Tshisekedi had received 38.5% of the vote and had been "provisionally declared the elected president".
The full results were, with turnout reportedly 48%
Analysis by Louise Dewast, BBC News, in Kinshasa
This is a historic moment for Tshisekedi's party, the UDPS, who have tried unsuccessfully to win power for decades.
But in recent days, rumours of a deal between Tshisekedi and Kabila have raised concerns among other members of the opposition. Mr Tshisekedi himself has acknowledged talks with the ruling party to prepare a transition of power.
Candidates can appeal results if they wish and the final decision will be announced by the constitutional court.
The big question now is how the country will react. There are serious concerns of unrest if people conclude that results have lacked in credibility.
The Catholic Church, who had the largest number of election observers deployed, has warned they will reject any results they believe to be fraudulent.
Presentational grey line
On Wednesday, anti-riot police were deployed in the capital, Kinshasa, amid fears that a disputed result could trigger violence.
Polling at the end of last month was marred by allegations of vote-rigging. The influential Catholic Church, which deployed 40,000 observers during the vote and has been monitoring the polls, warned that there had been irregularities.
The leaders of Zambia and South Africa urged officials to release the tally quickly to avoid growing suspicion around the poll, which has been marred by allegations of fraud.