With counting completed from nearly three-quarters of polling stations after last Sunday's poll, the "Yes" vote had won 83% and the "No" vote nearly 17%, the Independent Electoral Commission said.
The vote on whether to accept the proposed constitution was the first independent election to be held in the Congo for more than 40 years of dictatorship, wars and chaos.
Apollinaire Malu Malu, president of the commission, told reporters that the electoral body was still counting the remaining votes and said the Supreme Court would have to certify the final results and deal with any complaints. But as the results stood so far, the "Yes" vote lead seemed unassailable.
International observers judged the referendum free and fair. But the opposition has said there were irregularities and protested over the detention by police last week of a spokesman for the "No" campaign.
Joseph Kabila, who became president after his father, Laurent, was assassinated in 2001, had called for a big "Yes" vote for the constitution, saying its rejection would be disastrous for the country's peace process.
About 25 million people registered to vote in the referendum and for subsequent local, parliamentary and presidential elections, which under a peace deal must be held by next June.
Successful elections are seen as the key to consolidating peace in the mineral-rich country, which is the size of Western Europe but lacks even the most basic infrastructure in most areas.
Thousands of gunmen still terrorise civilians in the east and the United Nations has its biggest peacekeeping force in the Democratic Republic of Congo.