With most ballots counted, the ruling center-left Social Democrats had 28 percent of the vote, trailed by the Moderates with 19 percent and the nationalist Sweden Democrats with nearly 18 percent.
That could prove fatal for the Alliance, with the Liberal and Centre parties repeatedly ruling out a deal with "the devil", as Akesson occasionally calls himself.
That gave the centre-left 144 seats in the 349-seat parliament against 142 for the Alliance, suggesting weeks of uncertainty before a workable government can be formed.
"Sweden, birthplace of multiculturalism and model for the left, has finally chose to change after years of wild, uncontrolled immigration", Italian Deputy Premier and Interior Minister Matteo Salvini, leader of the far-right League, said in a statement Sunday night.
Sweden - which has a population of around 10 million - took in 163,000 migrants in 2015.
With most votes counted the biggest party (on 28 per cent) is still the ruling centre-left Social Democrats, the oldest and biggest party in Sweden, bleeding from the elderly working class part of its base but still a power.
Kristersson called on Lofven to resign, but rebuffed Akesson.
"That's not going to happen, that's not how it is", Akesson said.
"The Alliance will not govern or discuss how to form a government with the Sweden Democrats".More news: Call of Duty: Black Ops IV Blackout Trailer
Immigration has been a central issue of the campaign.
The Sweden Democrats, a party rooted in a neo-Nazi movement that has worked to soften its image, has played a role in breaking down longstanding taboos on what Swedes could say openly about immigration and integration without being shunned as racists.
With an eye on the European Parliament elections next year, Brussels policymakers are watching the Swedish vote closely, concerned that a nation with impeccable democratic credentials could add to the growing chorus of euroscepticism in the EU.
Mainstream politicians have so far ruled out cooperation with the Sweden Democrats.
The liberal-conservative Moderates came second with just under 20 per cent. SD won just under 18 per cent of the vote.
The latest opinion poll suggests that Prime Minister Stefan Lofven's ruling Social Democrats will substantially lose seats but still emerge a victor with an estimated 24.9 percent of the votes. "Historically it has been always been this way in Sweden", he said.
In Sweden, the speaker of parliament typically consults all party leaders after an election before tasking the one most likely to succeed at forming a government. "That's why the Sweden Democrats are as big as they seem to be", he told CNN in Stockholm. "I think that it shows that people are disappointed with how the country is run, and they are rooting for something, they want something different, but it's not necessarily the best kind of different", he said of the far right's rise. "What I can exclude is any direct or indirect co-operation with the Sweden Democrats", Interior Minister Anders Ygeman, a Social Democrat said.
Well aware that neither Lofven's "red-green" bloc nor his own Alliance had a chance of winning a majority, Kristersson has said Sweden needs "a strong cross-bloc cooperation to isolate the forces. pushing for Sweden to withdraw from worldwide cooperation".
These results forecast a hard transition period as parties work to form a coalition, which could take weeks. "It's time to take responsibility and talk to the Sweden Democrats", he said.