The opposition United National Movement (UNM) is in second place with about 27% of the votes.
Earlier, PM Giorgi Kvirikashvili claimed a "huge victory", but UNM leaders questioned that assessment.
Those statements were based on exit polls, which showed different results to the partial official ones.
More than 20 political parties competed for seats in the assembly in Saturday`s elections seen as a test of stability for the former Soviet republic.
Many voters had been undecided on the eve of the elections, according to opinion polls, with Georgian Dream and the UNM neck-and-neck.
They were weary about the elections, as poverty, unemployment and debt continue to blight their lives, the BBC`s Rayhan Demytrie in Tbilisi reports.
The exit polls also suggested that a pro-Russian party could enter parliament for the first time in Georgia`s post-Soviet history.
The Alliance of Patriots needs to secure 5% of votes, a requirement to be represented in the 150-member parliament.
Georgia briefly went to war with Russia in 2008 and wants to join Nato and the EU.
In a region of predominantly authoritarian states, it is seen a post-Soviet success story with a multi-party democracy, our correspondent says.
Voter turnout was low, with just 51% of those eligible casting ballots according to the election commission.
Mr Kvirikashvili praised the election as "truly free and fair" but some procedural violations were reported by observers.
A car bombing targeting a UNM lawmaker earlier this week in central Tbilisi saw the party accuse authorities of creating "a climate of hatred" before the polls.
A polling station was attacked Saturday in Marneuli, south of Tbilisi by a large group of UNM supporters according to police.
Georgian Dream had contested the election claiming modest economic success stories over the past four years - including a trade deal with Europe.
The UNM, which brought Georgia international recognition for its reforms under its flamboyant leader and former President Mikheil Saakashvili, was voted out in 2012 following a prison abuse scandal.
It hoped that Georgians had grown tired of the ruling party and its billionaire founder Bidzina Ivanishvili, our correspondent adds.
Despite having officially left politics, Mr Ivanishvili is still widely considered to be the country`s chief decision-maker.
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