The candidate from Mr Kabila's party had initially been expected to win, but ended up finishing third - and is not challenging the results.
The centre monitored the elections and has since announced that its tally differs from the official results as released by the election commission.
The result is certain to fuel further suspicion that Tshisekedi, who the last opinion polls before the election showed was well behind Fayulu, struck a power-sharing pact with Kabila.
Tshisekedi's victory was greeted by celebrations at the headquarters of the Union for Democracy and Social Progress (UDPS), the country's oldest and largest opposition party which was founded by his late father, Etienne, a former minister under the late dictator Mobutu Sese Seko who then became an opposition leader and a thorn in Mobutu's side for years.
"Results contrary to the truth of the ballot box will be considered as a constitutional coup", said a joint statement from the groups, which include Lucha (Struggle for Change) and Filimbi.
About 40% of DR Congo's population is Roman Catholic and the church has a wide network of schools and hospitals. The United States threatened sanctions against officials who rigged the vote.
UN Secretary General Antonio Guterres urged all sides "to refrain from violence and to channel any eventual electoral disputes through the established institutional mechanisms", his spokesman Stephane Dujarric said.
The largely peaceful election was marred by the malfunctioning of many voting machines that Congo used for the first time.
"Kabila will be able to influence Tshisekedi, who now owes his ascendancy to power to Kabila's control of the electoral commission", Mr. Besseling told AFP.
Defiantly, tens of thousands of voters in one of the barred communities held their own unofficial ballot on election day, and Fayulu won easily.
Many Congolese objected to Shadary, suspecting that Kabila would continue to rule from behind the scenes.
By breaking away from the opposition coalition supporting Fayulu, Tshisekedi "positioned himself to bargain with the regime", Englebert wrote.
Resident John Kabamba says it "may be a message that the publication (of the results) won't meet the expectations of the Congolese people".
Felix Tshisekedi was declared the victor of the country's long-delayed elections.
Pierre Englebert of the African Arguments newsletter said analysis of survey data "suggests that the probability Tshisekedi could have scored 38 percent in a free election is less than 0.0000". "This is the beginning of national reconciliation".
Now Congo faces a new leader who is little known after spending many years in Belgium and standing behind his outspoken father.
Election officials announced that opposition candidate Felix Tshisekedi won the election.
Over Caesar salad and iced tea, Trump projected confidence that his strategy was working, Fox News Channel's Bret Baier reported. Such an emergency declaration would represent a dramatic escalation of the dispute and would immediately draw legal challenges.
He may have a great offensive mind but there is little evidence he is ready to take over an National Football League franchise. Kingsbury has already spoken with with the Arizona Cardinals since USC chose to allow him to speak to professional teams.
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The New England Patriots have basically dominated the National Football League for the past two decades. It was the same guy that helped us get here. "It's a Bill Belichick-led team and defense", Rivers said.
And to me those are immediate trade. "We have firm confidence in the strong long-term fundamentals of the Chinese economy". Official U.S. fuel storage data from the Energy Information Administration (EIA) is due at 1800 GMT on Wednesday.
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