The 30 December poll was supposed to mark Congo's first uncontested democratic transfer of power in 59 years of restive independence and the beginning of a new era following 18 years of chaotic rule by President Joseph Kabila.
Its approval of the election's results are critical for the legitimacy of president-elect Tshisekedi's new government.
The move came as the pro-Kabila FCC coalition announced it had won a majority in legislative elections, which took place at the same time as last month's presidential poll.
Analysts said the Church's perspective would not go unnoticed.
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.More news: FDA inspections impacted by government shutdown
Corneille Nangaa, president of the CENI electoral commission, responded that the voting data could only be handed to the Constitutional Court in line with DR Congo legislation.
The court now has eight days to consider the request.
He has said that if Tshisekedi becomes president "he will be Mr. Kabila's stooge who will continue to pull the strings".
China and Russian Federation made clear that the top United Nations body should be concerned by the stability of the DR Congo as it undertakes what is set to be the first peaceful handover of power since its independence from Belgium in 1960. Several journalists were waiting for him as Tshisekedi has made no public appearances since the announcements by Fayulu and the Catholic Church that the figures giving him victory are not accurate.
"We don't expect the election to be annulled, but (a decision in favour of) a recount", Fayulu said.
A southern African regional bloc urged officials in the Democratic Republic of Congo Sunday to recount the votes of its recent contested presidential election.More news: Police respond to active shooting at New Jersey UPS facility
Fayulu, who has members of the Republican Guard deployed outside his home, called for a manual recount of the election. The Catholic observers are widely reported to have concluded that Mr. Fayulu was the victor.
Police patrols subsequently evicted protesters from surrounding streets using tear gas and rifle butts, leaving some 80 people injured.
What was the result of the presidential election?
The prime minister, in turn, must countersign presidential orders appointing or dismissing military chiefs, judges and heads of state-owned enterprises. The Church is highly trusted in the DRC and it is partly thanks to the feedback from its 40,000 election observers that there has been mounting pressure on Kabila regarding the outcome.
Most leaders issued statements calling for any disputes to be resolved peacefully, but notably lacking any congratulations for Tshisekedi.
Speaking by videoconference to the UN Security Council on Friday, CENI chief Nangaa appealed for "the new authorities to be supported by the global community".More news: SpaceX plans workforce reduction, some local employees sent home
At a special UN Security Council session on Congo Friday, the powerful CENCO Catholic bishops group, which enjoys widespread credibility across Congo, called on the worldwide body to ask the country's electoral commission to release the records of vote-counting at the polling stations to allow for verification. His attempts to cling to power after his term ended at the end of 2016 were arguably only thwarted by global and local opposition pressure, which met sometimes deadly resistance from Kabila's regime.