According to AFP, the West African country is known for its stable democracy, but tensions rose over Monday's presidential and legislative vote after Mahama claimed to have won a parliamentary majority and warned Akufo-Addo against stealing the vote.
The announcement was greeted with chanting and dancing by a crowd of supporters in the seaside capital Accra.
"Now is the time, irrespective of political affiliations, to unite, join hands and stand shoulder to shoulder," the 76-year-old president told excited supporters honking vuvuzuelas outside his Accra residence.
Mahama, 62, who has twice before lost to Akufo-Addo by a narrow margin, the last time in 2016 -- has yet to comment on the results.
The full count of the 275 parliamentary seats has not been announced and are expected to be very close between Akufo-Addo's centre-right New Patriotic Party (NPP) and Mahama's National Democratic Congress (NDC) party.
The country has recorded high levels of growth during Akufo-Addo's first term as he worked to persify an economy largely dependent on cocoa exports and more recently oil and gold.
"There could be a tendency for an incumbent, who has just secured a second term, to take it easy and relax," Akufo-Addo.
"I am of a different character. I give you my word that I will continue to work very hard to build a prosperous and progressive Ghana, for which we yearn."
Former Ghanaian president John Kufuor was one of the first to congratulate Akufo-Addo, saying on Twitter: "you deserve it."
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