Party, the Christian democratic Union, whose leader is Angela Merkel, won the local elections in North Rhine - Westphalia, where previously the authorities were social Democrats.
Projections based on partial counting showed Ms. Merkel's Christian Democratic Union (CDU) beating the Social Democrats by about 34% to a bit over 30.5%.
State polls have been seen as voter mood tests ahead of Germany's parliamentary elections in September.
Voters were choosing candidates for the state legislature, which may now change from its current SPD-Greens coalition.
In the state's last election in 2012, the Social Democrats beat the CDU by 39.1% to 26.3%.More news: Pablo Zabaleta relives favourite Manchester City moments after final home game
Party leader Martin Schulz had predicted a victory there would make him the next German chancellor - but the victory for Mrs Merkel's party will now give her a boost as she seeks a fourth term.
Germany's right-wing nationalist party, Alternative für Deutschland, won a little over 7% of Sunday's vote and will therefore have representation in the NRW parliament.
Following the result, Mr Laschet will discuss forming a coalition with the Free Democrats, which picked up 12.6% of the vote.
"My friend Emmanuel Macron was at the bottom five months ago, and now, he is president", said Schulz of the new French leader who took office on Sunday. But overshadowing Kraft's campaign for yesterday's election has been criticism of her government's handling of the 2016 New Year's Eve attacks on women in Cologne, as well a December terrorist attack on a Berlin Christmas market that was carried out by a one-time resident of North Rhine Westphalia.
"This is a hard day for social democracy", said Martin Schulz, visibly shaken.More news: Trump Defends Sharing 'Facts' About 'Terrorism And Airline Flight Safety' With Russia
Schulz defiantly asserted that the SPD is a party that knows how to fight back and noted, "The federal elections are still a long way off". The opposition Left Party fell just short of the 5% needed to win seats. It is home to almost a quarter of all German voters - more than 13 million.
He urged the party to concentrate now on the national election, saying: "We will sharpen our profile further - we have to as well. I am convinced that for the past seven years, step by step, we have helped this state move forward", Kraft said in the state capital, Düsseldorf.
In the state of North Rhine-Westphalia, there was no "Schulz effect", and his personal image is being tarnished following three successive defeats in the Saarland, Schleswig-Holstein, and now North Rhine-Westphalia.
Carsten Nickel, an analyst with Teneo Intelligence, said the NRW result was a sign that Merkel was on her way to re-election. It carried even higher stakes this year, being the last regional vote before the national election and having a direct impact on the SPD's ability to close a nationwide gap of around 10 percentage points with the CDU.
The AfD is now represented in 13 states across Germany.More news: Ford to cut 1400 salaried jobs in North America, Asia
According to Spiegel Online magazine, the victory in North Rhine-Westphalia is a great success for Merkel. We had two goals in this election, to stop red-green and become strongest political power. SPD state premier Hannelore Kraft quickly conceded defeat and said she would stand down as her party's local leader.
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