"This could be the beginning of a historic era in the (Democratic Republic of the Congo) if peace holds and finally the election results are accepted", said CBS News' Debora Patta.
There are widespread concerns that a contested result could spark unrest.
Leading opposition candidate Martin Fayulu has urged the electoral commission to announce the true results as quickly as possible and warned it not to "play with fire, it is very unsafe".
The South African government has tweeted a photo of Mr. Ramaphosa's meeting with President Lungu.
Many Congolese objected to Shadary, suspecting that Kabila would continue to rule from behind the scenes.
Felix Tshisekedi, the leader of the Democratic Republic of the Congo's main opposition party, has been declared the surprise victor of the 30 December presidential election in the vast central African country.
At stake is political stewardship of the notoriously unstable central African nation which has never known a peaceful transition of power since independence from Belgium in 1960.
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Just hours later, the Church said election results tallied by its 40,000 observers scattered across the country showed a different victor, without specifying who.
Congo on December 30 voted for a successor to departing President Joseph Kabila.
The church's long-awaited announcement, hours after the official results, brought fresh uncertainty.
Riot police were deployed outside the offices of the DRC's election commission and elsewhere in the capital, Kinshasa.
The other main opposition candidate, former oil executive Martin Fayulu - who was tipped as the favourite in the few pre-election opinion polls - also sounded a conciliatory tone on Tuesday.
Felix Tshisekedi, leader of the Congolese main opposition party, the Union for Democracy and Social Progress (UDPS) who was announced as the victor of the presidential elections gestures to his supporters in Kinshasa, Democratic Republic of Congo.
The electoral commission started deliberating and evaluating presidential election results on Tuesday evening, after which it will announce the victor. He said the delay is to "fudge the results" and warned that his coalition would release its own figures if the official ones are in doubt. The constitutional court then has seven days to consider disputes before results are final.
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The influential Catholic Church notably said the result "does not correspond" with the data collected from polling stations and counting centres by its 40,000 election monitors, although it did not identify the person who it believes has won. He is set to replace President Joseph Kabila, who has ruled the Democratic Republic of the Congo for 18 years.
It "may be a message that the publication (of the results) won't meet the expectations of the Congolese people", resident John Kabamba said.
They would be deployed if needed to protect USA citizens and diplomatic facilities in DR Congo's capital, Kinshasa, he said.
Fayulu alleges that Kabila engineered a backroom deal with the largely untested Tshisekedi to protect his power base in a country with staggering mineral wealth.
Even before the announcement, some observers suggested that Kabila's government made a deal with Tshisekedi as hopes faded for ruling party candidate Emmanuel Ramazani Shadary, who received just 23 percent of the official results. Joseph took over in 2001 when Laurent was assassinated.
Congolese activist groups are urging people to "be ready to massively take to the streets" if the electoral commission does not publish presidential election results in accordance with "the truth of the ballot boxes".
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