Australian authorities picked up the group's yacht after receiving a distress call earlier this week.
It is believed the group left Malaysia on a yacht bound for New Zealand last month.
Mr Bowen said the group had decided to seek asylum in Australia after talks with immigration officers on Wednesday.
'I honestly think that's a good outcome, it means they won't be undertaking, yet again, another further dangerous boat journey,' he told ABC Television.
The minister said the group's initial claims would be processed in Darwin.
Eight of the ten asylum seekers had valid passports, which made a difference because their identities could be established, he said.
'Where someone arrives with a passport and we assess that passport as being valid and legitimate and not a forgery, it does make the rest of the process easier.'
The group intended to go to New Zealand to claim asylum because there is no mandatory detention.
Mr Bowen said NZ authorities had warned the group of the perilous journey they risked across the Tasman.
The men, women and children say they are members of Falun Gong, a spiritual group, whose members claim to be persecuted by Chinese communist leaders.
The group will be housed at detention facilities in Darwin.
The Greens said the case highlighted the different way Australia treated people who arrived by boat and those who arrived by plane, but also wanted to be granted asylum.
'Those who arrive by plane and seek asylum can apply for their protection living in the community,' Senator Sarah Hanson-Young said.
'Those who arrive by boat we lock up, we throw away the key.
'This is a system that needs to be rebalanced.'