Fresh elections being held three months after a disputed poll led to violent protests.
The institute, a grouping of think tanks and polling organisations, said it interviewed about 17,000 voters in 200 polling stations around Moldova, and that the poll had a margin of error of plus or minus two percentage points.
If official results confirm the projections, the opposition parties would be in a position to form a government if they forge an alliance.
The results will also determine who succeeds Vladimir Voronin, Moldova’s president, who has ruled the former Soviet republic since 2001.
Elections in April, in which the Communists won about 50 per cent of the vote, triggered huge street protests and riots, which left at least two people dead.
Voronin called new elections in June, after politicians failed to elect a new president due to a boycott by the liberal opposition parties.
Ten parties took part in the latest elections, with 2,000 polling stations open for the country’s 2.6 million eligible voters.
About 3,000 foreign and Moldovan observers monitored Wednesday’s vote, in an effort to discourage any attempts at election fraud.
The result of the vote will significantly affect how the country steers its foreign policy towards Russia and the European Union.