The final tally was due to be released sometime on Thursday.
The new law addresses corruption, political patronage, land grabbing and tribalism, which have plagued Kenya since it won independence in 1963.
Andrew Simmons, Al Jazeera's correspondent in Nairobi, said this was a victory for all Kenyans.
"It is really a case whereby not only can the "yes campaigners" claim victory, but Kenyans as a whole can celebrate a victory of common sense and a political maturity," he said.
"No one would compare this directly with the 2007 hotly contested elections, but it was a test case because there was a lot of resistance to this constitution and there were fears there would be violence.
"The voting passed by peacefully and the results it would appear would seem likely that there would not be any violence."
Kenya's electoral authority said on Tuesday the process would be more transparent than the 2007 national election, which was marred by claims of fraud in favour of president Mwai Kibaki. Those allegations led to violence in which at least 1,300 people were killed.
Kibaki assured the country that security had been strengthened at polling stations to prevent any violence.
Several hundred people in the country's Rift Valley fled their homes before the vote, fearing a flare-up of violence. The valley is home to a large concentration of "no" voters.
It was also the site of some of the worst attacks during the last election. Tribesmen used bows and arrows to fight each other, gangs hacked opponents to death and police were accused of shooting sprees.
The new charter was a key provision of the power-sharing deal struck in 2008 between Kibaki and Odinga.
William Ruto, Odinga's former ally and a cabinet minister based in the Rift Valley, is spearheading the "no" campaign, which is angry with provisions related to land ownership.
Kibaki and Odinga both back the "yes" campaign, bringing two of the country's major ethnic groups onto the same side.
The draft constitution cuts down the president's enormous powers by setting up an American-style presidential system of checks and balances, part of the reason the draft appears to have wide support.