Stockholm, Sweden (CNN) -- Sweden's ruling center-right coalition won re-election Sunday, marking a historic moment as a non-socialist government was elected to a second term for the first time in the country's political history.

Swedish Prime Minister Fredrik Reinfeldt's center-right four-party coalition -- made up of the Moderates, the Liberals, the Christian Democrats and the Centre party -- held on to power, but lost its outright majority.

"The Swedish people have cast their vote, and they have ruled that we are the ones who should keep governing," Reinfeldt said at his party's election night celebration.

His coalition won 49.3 percent of the vote, officials at the Swedish Election Authority said after all 5,668 voting districts reported. The opposition "red-green" coalition -- consisting of the Social Democrats, the Left party and the Green Party -- had 43.7 percent of the vote, election officials said.

The far-right anti-immigration Sweden Democrats party also made a strong showing, winning 5.7 percent of the vote and a place in the national parliament for the first time. With possession of 20 seats, the party could wind up tipping the balance of power between the two major coalitions, although party leaders, including Reinfeldt, have vowed not to cooperate with the Sweden Democrats.

The leader of Sweden's red-green opposition coalition, Mona Sahlin, conceded defeat Sunday night. She told her supporters they were not able to regain the trust of the voters.

"We have lost," she said, stressing that the center-right coalition also failed to get an outright majority.

The ruling coalition won 172 seats, while Sahlin's group took 157 in the 349-seat parliament.

Sweden has a long tradition of socialist rule, with a cradle-to-grave welfare system. But the global financial crisis threw Sweden into one of its worst economic downturns since World War II.

The ruling conservative coalition, which came into power in 2006, imposed a string of austerity measures and managed to turn Sweden's economy into one of the strongest in Europe, with an expected growth of 4.5 percent this year. The crisis management appears to have impacted many voters.

"I think the economy is the key issue," said one man at a Stockholm polling station. "I think Sweden has done very well for the last few years during the global financial crisis, and I hope the government will stay on."

But with a tightening of fiscal policy, several groups in Swedish society have seen their situation worsen. Pensioners and sick people are among the hardest hit, and the leader of the red-green coalition had urged voters to vote for change on Sunday.

"There is a clear difference between the left's and the right's tax policies towards working people and pensioners," said one elderly woman who had just cast her ballot on Sunday. "My pension has gone down during these last years."

"The moderate party and the center-right alliance seeks the confidence of the voters," Reinfeldt said in a televised speech on Saturday, the eve of the election. "We do this with a promise to take responsibility. We have taken Sweden through a difficult economic crisis. Many decisions have been hard to make, and not everything has been right from the beginning."

But, he said, "after a difficult financial crisis, confidence in the future is now growing in our country. It is great to see how Sweden gets back on its feet. We are seeing more jobs and the unemployment is going down. Sweden today has Europe's strongest economy, but there is a risk for new troubled times. There are countries in our surroundings that have lost control over their economy and have had to make hard cuts and increase taxes. This will always hit the weakest the hardest. Don't put Sweden in this situation."

Meanwhile, Sahlin said, nearly all Swedes want "a health care based on their needs, not their wallet, and a school that helps all children gain knowledge, regardless of their background ... I want to take responsibility for Sweden, the welfare state. If we can handle the jobs situation, then our economy will grow, and we can impose our welfare."

"I am for reductions in tax but not at any cost," she said. "Don't vote away Sweden the welfare state. What we sell and tear down now will never come back."

The far-right Sweden Democrats, which received 2.9 percent of votes in 2006, nearly doubled its votes this year. But its anti-immigration policies have caused all the main party leaders to vow not to cooperate with it, even as it won seats.

"I think it is more important than ever that everyone goes to vote today, so that we can stop them," one young woman voter said, referring to the Sweden Democrats. "I think it would be a day of shame for all Swedes if those people would come into parliament."