Earlier a woman escaped from the house unharmed before a policeman carried her to safety while other officers protected them with shields.
A police spokesman said she was the ex-wife of the armed man, a former member of the Yamaguchi-gumi, Japan's largest crime syndicate.
One of the police officers died several hours after he was shot on Thursday evening while helping his colleague who had been hit earlier.
The rescued officer is in stable condition while the two children have been taken to hospital in the city of Nagoya.
About 170 police officers surrounded the house on Friday morning and state broadcaster NHK reported that schools in the area were closed for the day.
Police said the man's wounded daughter had been co-operating with them and had spoke to her father several times to urge him to surrender.
|Recent shootings have drawn attention to the|
yakuza organised crime syndicates [Reuters]
The standoff comes a month after two incidents which shocked a country where gun control is tough and shootings are rare.
Last month, a gangster shot a fellow mobster in a Tokyo suburb and hid in an apartment before shooting himself, while another gangster shot dead the mayor of Nagasaki.
Kensei Mizote, the minister in charge of the police, called for tough measures.
"We must push ahead with effective measures to control firearms, including steps against gangsters, so that such an incident won't happen again," he said.
Gun-related crimes in Japan hit a record low of 53 last year, resulting in only two deaths out of 36 gang-related shootings, mostly involving organised crime members.
Last month's shootings prompted calls for even stiffer gun control in a country where legal firearms are mostly in the hands of police and hunters.
Illegal guns tend to be owned by members of the yakuza, Japan's crime syndicates whose mainstays include prostitution, drugs, extortion and even finance operations.
The recent shootings have revived international interest in the yakuza, whose members have been known for their elaborate tattoos and missing little fingers – cut off to apologise for mistakes or show loyalty to the boss.
There were 41,500 yakuza members in 2006, slightly lower than 2005. Japan does not outlaw gang membership.