The Social Democrat Siumut party will form a government with the centre-right Democrats and the anti-secessionist Atassut, newspaper Sermitsiaq reported.
A snap election, held on Friday, was called in October after another Siumut-led coalition collapsed when it was revealed that former prime minister Aleqa Hammond had used public funds to cover private expenses.
But the scandal only had a limited impact on Siumut, which has ruled Greenland for all but four years since the granting of home rule in 1979, and the party won 34.3 percent of the votes cast — just slightly ahead of the rival, leftist Inuyt Ataqatigiit (IA) with 33.2 percent.
Both parties won 11 seats in the 31-member parliament, and with a total of 17 seats the new government will only have a narrow majority.
As part of its plans to achieve economic independence from Denmark the Hammond government lifted a ban on mining uranium, which IA wanted to reinstate and had promised to put to a referendum.
However, after it emerged that Greenland's undisputed mineral wealth will take longer to develop than initially thought, politicians have focused on other parts of the economy, which is dominated by fisheries and an annual subsidy from one-time colonial master Denmark.
Kielsen is a former police officer who built his campaign on tackling political corruption and nepotism in the vast island with a population of just 57,000, and on pledges to develop the fisheries sector.