President Kagame has yet to announce whether he will seek another term, but he is widely expected to do so.
The vote took place amid criticism from donor countries including the United States, human rights groups and the country's only opposition party, the Green Party.
"The election did not guarantee a level playing field since the Cabinet issued a one-week short notice for the referendum to be conducted," the Green Party said on its official website Saturday. Earlier, the European Union Delegation to Rwanda had also expressed concern about the short notice of the referendum.
National leaders across Africa have turned their eyes toward tiny Rwanda this week as voters passed judgment on the constitutional change.
It's not as if Kagame just assumed the position. He has already been president for 15 years, having taken office in 2000.
More than 6 million Rwandans were expected to vote on Friday.
But it is not so much the results that other leaders around Africa will be watching. The extension of term limits was virtually a foregone conclusion. Voters -- some enthusiastic, some intimidated -- were expected to approve the extension.
What the other leaders will watch, said Omar McDoom, an assistant professor in comparative politics at the London School of Economics and Political Science, will be the reaction of the international community, and in particular of donor countries, who hold great influence in Africa.
If the donor community takes no action, other leaders will feel free to ignore term limits with impunity, McDoom said.
A trend across Africa to hang onto power
The desire to hang onto office for decades runs strong across the continent despite opposition.
"The decision of leaders to hang onto power stands as a challenge to the prescription of the African Union" regarding democracy and good governance, said David Zounmenou, a senior researcher at the Institute of Security Studies in Pretoria, South Africa.
"It's a tricky game for African leaders," Zounmenou said. "They are watching closely what is happening."
Always in the background in Rwanda -- and in neighboring Burundi -- is the possibility of ethnic violence. Both nations, which are small, landlocked countries in East Africa, have populations that are majority Hutu and minority Tutsi.
In 1994 in Rwanda, extremist Hutus slaughtered an estimated 800,000 Tutsis and moderate Hutus in the space of about 100 days.
Kagame led the rebel force that ended the genocide.
To those who argue that classic Western democracy does not work everywhere, Rwanda is Exhibit A. In some countries, elections are not about issues such as whether to spend more money on education or defense, but resemble a census to determine which ethnic group shall govern.
Burundi 'almost on the verge of civil war'
Kagame's Tutsi-led government has intimidated the opposition and stifled the independent press. To those who would argue for freer, more competitive elections, many Tutsis would reply that they tried that and they got killed by the hundreds of thousands.
Rwanda's population is 84% Hutu and 15% Tutsi.
Kagame, the Tutsi President, was expected to win the referendum, despite the Hutu majority, partly because he has led the country during a period of relative prosperity, but equally because his intelligence network and his suppression of free competition have created a "culture of anxiety," McDoom said.
But the country's Justice Ministry rejected that analysis.
"The purpose of democracy in Rwanda is to enable people to play a central role in determining their own destiny," the ministry said Friday. "Today's referendum was a critical moment in our country's history as it marks the culmination of a long constitutional process initiated and led by the people themselves. This process demonstrates the strength of our democracy and is representative of the will of the people."
Zounmenou said that, while the ethnic makeup of Rwanda's neighbor, Burundi, is similar -- and Burundian Presdient Pierre Nkurunziza has recently been elected to a third term viewed by many as unconstitutional -- the situations are different.
"In Burundi, they are almost on the verge of civil war," Zounmenou said. "In Burundi, this may go differently than in Rwanda because of the hold that Kagame has on the people and the fear that people have for him."
Experts agree that what happens after the results of Friday's referendum are in will reverberate far beyond the borders of tiny Rwanda.
"What the donor and international communities do in response to this will send a very powerful signal to other leaders whose second term is coming to an end," said McDoom.