According to Mr. Jean Yado Toe, official with the Independent National Electoral Commission preliminary results in Burkina Faso's legislative and municipal polls will be due by Friday, adding that counting of votes is finished in almost all polling stations, on Monday.
He said regional results were already being tallied and that preliminary results were likely on Thursday or Friday.
Mr. Toe said the voting had been largely peaceful, but acknowledged that there were some incidents involving destroyed ballot boxes and insufficient voting papers.
The vote was the last major test of Compaore's regime before a presidential election in 2015 in the landlocked West African country.
Some of President Compaore's supporters want him to change the constitution so he can stand again in three years' time, when his mandate is due to expire.
The powerful ruling Congress for Democracy and Progress (CDP) party is expected to easily win Sunday's elections.
The head of the opposition in the outgoing parliament, Mr. Benewende Stanislas Sankara, also hailed the calm vote and the high turnout, stating that "The essential thing is that democracy should emerge strengthened and that we now set our sights on uprooting the regime of Blaise Compaore from the presidential palace".