"They called for the change of the constitution and they have made it," he said, according to the pro-government New Times newspaper.
Kagame, 58, could now be in power potentially for another 17 years.
"What is happening is the people`s choice," he told reporters after casting his vote on Friday.
The amendment allows Kagame to run for a third seven-year term in 2017, at the end of which the new rules take effect and he will be eligible to run for a further two five-year terms.
The proposed amendments have been denounced by Washington and Brussels as undermining democracy in the central African country.
"Yes, Rwandans have spoken out loud and clear; they have taken their own destiny and that of their country into their own hands," a New Times editorial said on Saturday.
"There was no doubt from the beginning that the results from the referendum to amend the constitution was what was expected, regardless of the lame attempts to pour water on the process by Rwanda`s closest `development partners`," it said in an allusion to US and European criticism.
Kagame has run Rwanda since his ethnic Tutsi rebel army, the Rwandan Patriotic Front (RPF), ended a 1994 genocide by extremists from the Hutu majority, when an estimated 800,000 people were massacred, the vast majority of them Tutsis.
Changes rushed through
The issue of long-serving rulers clinging to power has caused turmoil in Africa, where some leaders have been at the helm for decades.
The amendments brought criticism from the United States and the European Union as well as the country`s tiny opposition Green Party.
On Friday, the EU delegation in Kigali said there had been a lack of "sufficient time and space for debate" on the issue, with the date for the referendum only announced on December 8 and the draft of the changes "only published publically less than one day ahead of the vote."
The country`s tiny opposition Green Party also protested it was impossible to organise a counter campaign at such short notice.
"The opposition would have won," Green party president Frank Habineza said in a statement on Saturday.
"The Democratic Green Party of Rwanda will not give up on the struggle to make Rwanda a vibrant democracy."
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