If the results are confirmed, Abdel Aziz - who led coup last year and resigned from the army to contest the elections - will become president without the need for a runoff.
Abdel Aziz deposed Sidi Ould Cheikh Abdallahi as president in August, provoking international criticism.
He has promised cuts in food and fuel prices that are likely to endear him to Mauritanians, 40 per cent of whom live under the poverty line.
Abdel Aziz's closest challenger is Messaoud Ould Boulkheir, parliamentary speaker and candidate of an anti-coup front, with 16.63 per cent of votes, the government figures show.
In third place is Ahmed Ould Daddah, the head of the main opposition party, with 13.89 per cent.
The electoral commission said voter turnout was at 61 per cent.
Saturday's elections are intended to restore constitutional democracy to this arid but potentially oil-rich country in northwest Africa.
Some 1.2 million of the nation's three million people were eligible to vote in the polls, which were monitored by international observers from the African Union, the Arab League and the association of Francophone countries.
The election had been put off for a month in order to end an opposition boycott.