Mr Hatoyama's Democratic Party of Japan ousted the long-ruling Liberal Democratic Party in an election last month.
He faces pressure to make good quickly on campaign promises to focus spending on consumers, cut waste and reduce bureaucrat control over policy.
62-year-old Mr Hatoyama stood and bowed after the vote.
'I have mixed feelings of excitement about changing history and the very heavy responsibility of making history,' Mr Hatoyama said.
'The fight begins now,' he said as he left home earlier.
Mr Hatoyama's cabinet, a delicate balance of former Liberal Democrats, ex-socialists and younger conservatives, must hit the ground running.
His choice of 77-year-old Hirohisa Fujii as finance minister has soothed some analysts' concerns that the government's spending programmes will inflate an already huge public debt.
Japan is struggling to emerge from recession and cope with the cost of a rapidly ageing society.
Mr Fujii moved currency markets even before he was sworn in, pushing the yen up 0.5% against the dollar by telling reporters a strong yen had merits for Japan's economy and that recent moves were not rapid.
Mr Hatoyama's choice of Shizuka Kamei, the outspoken head of a coalition partner, as minister for banking supervision and postal services has also spooked some experts worried about Mr Kamei's opposition to market-friendly reforms.
Mr Hatoyama's vow to steer Japan on a more independent diplomatic course has sparked concerns about possible friction with top ally the US ahead of his diplomatic debut there next week, where he will meet President Barack Obama.
The US-educated Mr Hatoyama is expected to reassure Mr Obama over ties and perhaps postpone calls for re-negotiation of agreements on US troops stationed in Japan.
