But the results declared Tshisekedi the victor with 38.57 percent, while Fayulu came second with 34.8 percent.
Democratic Republic of the Congo has been in the grip of a two-year political crisis triggered by Kabila's refusal to step down when his two-term constitutional limit expired at the end of 2016.
The outfit already exceeds the threshold of 250 seats needed for a majority out of a total of 500, according to an AFP count based on results released by the Independent Electoral Commission (CENI).
"Between them, they didn't get more than 40 percent".
The turmoil has darkened hopes that the country will have its first peaceful handover of power since it gained independence in 1960.
Martin Fayulu has denounced Tshisekedi's victory and apparent power-sharing deal with Kabila as an "electoral coup".
The last two elections in 2006 and 2011, both of which were won by Kabila, were marred by bloodshed, and many fear a repeat of the violence if the result lacks credibility.
The Catholic Church in the Democratic Republic of Congo (DRC) also challenged the results published by CENI.
Ahead of the news conference, police roughly dispersed a crowd of around 150 Fayulu supporters who had gathered near the court, and also saw off others who had rallied near his Kinshasa home.
Pro-Kabila candidates had secured 288 of the 429 seats so far declared, with 141 going to the opposition.
"I demand a hand recount of all votes for the three elections (presidential, national legislative and provincial)".
Worldwide observers have been closely watching developments in sub-Saharan Africa's largest country, which covers an area equivalent to that of western Europe, with reactions to the election outcome guarded.
Most leaders issued statements calling for any disputes to be resolved peacefully, but notably lacking any congratulations for Tshisekedi.
Pre-election polls indicated that Fayulu wasthe favorite to replace outgoing President Joseph Kabila.
Speaking by videoconference to the UN Security Council on Friday, CENI chief Nangaa appealed for "the new authorities to be supported by the worldwide community".
But Bishop Marcel Utembi, head of CENCO, which represents DR Congo's Catholic bishops, urged the Security Council to ask the election committee to release data on the counting at polling stations to allow for verification.
In turn, Kabila - who has ruled the country with an iron fist since 2001 - was likely to seek immunity from prosecution and protection for his family's business assets.
The authorities late Friday imposed an 8pm-5am curfew in Kikwit, a Fayulu stronghold in the west of the country, city mayor Leonard Mutangu told AFP.
In the eastern city of Goma, at least one person was killed, officials said.