The ballots are still being counted a day after the polls were held, with preliminary results to be announced on Monday, Iraq's electoral commission said.
Nearly 80 per cent of the region's 2.5 million eligible voters turned out for the dual elections, a full six months after the rest of the country held provincial elections.
Five candidates registered for the presidential race, while 24 political lists were competing for the 111 seats in the regional assembly.
Massoud Barzani, the incumbent regional president, was widely expected to be re-elected and his Kurdistan Democratic Party (KDP), along with the Patriotic Union of Kurdistan (PUK), headed by Jalal Talabani, the Iraqi federal president, had been considered likely to sweep the parliamentary poll.
The two factions have dominated the region's politics for decades and have been in power since 1991 when the region became semi-autonomous.
But a win by Goran in Sulaimaniya could strike a blow to the ruling coalition.
Al Jazeera's Hoda Abdel Hamid, reporting from Irbil, the regional capital, said defeat for either of the giants of Kurdish politics would be highly symbolic.
"Jalal Talabani and Massoud Barzani are really considered the fathers of the Kurdish region, in the sense that they are the ones that led the struggle against what was perceived as the oppressive regime of Saddam Hussein," she said.
"They are the ones who got the autonomy for this region and for ... the first time since then they have been defied and one of them lost."
Election officials in the region hailed the vote as transparent, but opposition parties complained of instances of fraud.
A spokesman for the Goran party said fraud was especially "significant" in the extended hour of voting.
"Tens of thousands of security forces voted today [Saturday] when they should have voted on the 'Special Day'", he said, referring to Thursday, when Kurdish members of Iraq's armed forces were to cast their ballots.
"Many others who could not find their names on the registration lists, voted. We accuse the Iraqi high electoral commission of being biased and of doing nothing.
However, the ruling Kurdistan Democratic Party (KDP) dismissed the allegations of fraud as "completely baseless".