Official results are expected early on Monday, but Mr Aso conceded the LDP was heading for a big defeat.
"These results are very severe," he said at party headquarters. "There has been a deep dissatisfaction with our party."
The LDP's Kotaro Tamura said: "We made too many mistakes. Very crucial mistakes... we changed prime minister three times without holding an election."
Mr Hatoyama, who is almost certain to lead the next government, is the wealthy grandson of the founder of Bridgestone tyres. His other grandfather was a former LDP prime minister.
He has promised to boost welfare, reform the bureaucracy and seek a more balanced relationship with the United States.
Alastair Leithead, BBC News, Tokyo
It's a massive swing. What the opposition can do now they are coming into power, and untested, is deal with the serious problem revolving around the economy and the recession.
Unemployment is at the highest level it ever has been and by the end of next year Japan will no longer be the second biggest economy in the world - that will be China.
Almost a third of the people here will be pensioners and therefore there will be fewer taxes coming in, more money going out.
It's a very difficult position that Japan is in. People have voted out a party that was in power almost without break for 50 years.
They are now looking to a new and inexperienced government to try and deal with some difficult challenges.
He said after polls closed: "We will not be arrogant and we will listen to the people."
But the BBC's Roland Buerk in Tokyo says Mr Hatoyama will have little time to savour his victory. Elections to the less powerful upper house are due next year, he says, and the DPJ will want to retain its control there to push through its agenda.
Turnout in Sunday's election was reportedly just under 50%, slightly down from 2005 when elections saw the charismatic Junichiro Koizumi's LDP elected with a significant majority.
Officials said the turnout held up despite a combination of typhoon-triggered rainfall around Tokyo and a government warning that a swine flu epidemic was under way.
Japanese broadcaster NHK announced its exit polls moments after voting ended at 2000 (1100 GMT), saying they showed a major power shift in Japan.
"Our exit polls show the main opposition Democratic Party will seize more than 300 seats, way more than a majority in the lower house," said the newsreader.
EXIT POLL PROJECTIONS
National broadcaster NHK: DPJ 298-329 seats; LDP 84-131
Private network TV Asahi: DPJ 315 seats
Tokyo Broadcasting System: DPJ 321 seats
Nippon Television: DPJ 324
Outgoing 480-seat lower house of parliament: LDP 303; DPJ 112
"That signals a defeat for the governing coalition."
The LDP had 303 seats in the outgoing parliament, compared to the DPJ's 112. The projections were based on exit polls of roughly 400,000 voters.
If the DPJ were to gain such a landslide majority, it could establish a new cabinet within the next few weeks.
It won control of the upper house in July 2007, amid voters' anger at a series of scandals and the loss of millions of pension payment records.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.