Angela Merkel was basking in a historic third-term election victory in Germany on Sunday night, having led her conservatives to their best result in more than 20 years.

Merkel's Christian Democratic Union and its sister party won 41.5% of the vote, with analysts calling the win a personal victory for the 59-year-old, who is now on track to overtake Margaret Thatcher as Europe's longest-serving female leader.

Merkel's performance was compared to that of conservative chancellor Konrad Adenauer, who was the last chancellor to secure a Bundestag majority without need of a coalition partner since 1957. After a campaign that concentrated almost solely on Merkel's personality and solid leadership in times of economic turmoil but was thin on detailed policy, she came within a whisker of obtaining an absolute majority, falling just five seats short.

Final results gave the CDU/CSU 311 seats, the Social Democrats 192, the Left party 64 and the Greens 63.

The historical dimensions of the election were clear, with Merkel set to become just the third postwar chancellor to secure three election wins, after Adenauer and Helmut Kohl, who brought her into the party as an inexperienced and gauche 35-year-old.

She has also bucked the European trend by becoming the only leader in the eurozone, whether from left or right, to be re-elected since the snowballing of the euro crisis in 2010. Out of 17 countries in the eurozone, 12 governments have fallen, indicating how protected under Merkel's leadership Germans feel from the crisis.

In a result that was closely watched across Europe, Merkel crushed her opponents – and, indeed, some of her allies.

Her coalition partner, the FDP, fell out of parliament for the first time since it was formed after the second world war, securing just 4.8% of the vote. All other parties – with the notable exception of the eurosceptic Alternative für Deutschland – lost ground.

The French president, François Hollande, was the first world leader to offer his congratulations, but the wider implications for Europe, austerity, the euro crisis and David Cameron's hopes of repatriating powers from Brussels were less clear.

Merkel will still have to rely on a coalition partner for a secure governing majority. Without her former liberal allies, she might now have to turn to the SPD, which is firmly opposed to Cameron's ideas of wresting powers back from the European Union.

The scientist and pastor's daughter who grew up in communist East Germany appeared to her party faithful in Berlin just 45 minutes after the first exit polls were released, a clear indication of the confidence there was in the win.

"We can surely celebrate this evening," she said, beaming at the largely young crowds who chanted "Angie, Angie". "This is a super result," she told the party faithful.

After thanking her campaign team in a rare emotional moment, she then turned to her husband, the chemistry professor Joachim Sauer, who was standing among the crowd, and said: "And of course my husband, standing on the sidelines, who has had to put up with quite a lot." Sauer smiled shyly back at her.

Merkel's official biographer, Stefan Kornelius, told the Guardian: "This usually distant and unemotional women is grinning and cheering all evening. Finally she gets the reward she was denied for two consecutive elections. But she knows the traps. Certainly her party is that close to an overall majority – the first time since Adenauer in 1957.

"But waking up tomorrow morning with all votes counted she might need to find a coalition partner anyway."

While Merkel's CDU celebrated its historic victory, the centre-right Free Democrats were contemplating the worst result in their 65-year history after failing to reach the 5% threshold necessary to enter parliament. The radical Left party (8.6%) was celebrating what appeared to be its new position as the third biggest force in the Bundestag. The other remarkable breakthrough of the evening was the sudden emergence of the eurosceptic AfD as a force to be reckoned with.

The party was just 0.3% short of the necessary threshold to secure Bundestag representation for the first time since its formation. It had promised that if it did so, it would change the terms of the euro crisis debate in which Germany has repeatedly sanctioned bailouts for countries in fiscal difficulties.

"We have to rethink the euro crisis," said Frauke Petry, an AfD leader. "We have to allow weaker countries like Greece and Spain and Portugal to leave the euro and rebuild their economies and then maybe return. We don't think we should pay for debts that have been accumulated by these countries. We think we will be able to push CDU and SPD towards new positions. Many members of the CDU quite agree with us but haven't said so in public."

Despite the record victory, Merkel might struggle in her third term. She could yet be forced into a coalition, most likely with the Social Democrats (SPD), who, with 25.7%, secured the second worst result in their history, or even the Green party (8.4%), with whom the CDU has entered government on a state level.

Otherwise she could find herself struggling to push legislation through both chambers of parliament, with the upper house dominated by left-leaning parties.

While the issue of the euro crisis played a minimal role in the election campaign, it is likely to take on more of a prominent role now that Merkel, praised and criticised in equal measure for her handling of the crisis, has been endorsed for a third four-year term. Wolfgang Schäuble, the finance minister and close Merkel ally, told German television in an interview that the conservatives' win should reassure Europeans. "Europe doesn't need to worry about the German elections," he said.

"We will remain reliable in the role of stability anchor and the growth motor of Europe … Germany continues to have an important leadership responsibility".

• This article was amended on 23 September 2013. The original version stated that the result was the worst for the Free Democrats in their 75-year history; in fact the party was founded 65 years ago.