Logie wil be an under-secretary, typically a job which focuses on a specific project.
The four roles are outside Cabinet as part of the Greens' support deal with Labour.
Gareth Hughes will be the party's chief musterer (whip) and Marama Davidson will be deputy musterer.
Shaw said their priorities would remain the same.
"The Green Party's ministers will work hard alongside our team and the rest of the Labour-led government to ensure we make a positive contribution to the lives of New Zealanders by addressing our priority areas of climate change, clean water, and ending poverty."
Shaw told Newstalk ZB he was not permitted to state who had which roles.
"The Prime Minister has that prerogative and I understand that she's intending to make an announcement about the full Cabinet line-up including Labour, New Zealand First and Green ministers and under-secretaries next week," he said.
"It's a very exciting day for us.
"We are very happy with the roles the Green Party have been assigned. They are completely consistent with what we said during the election campaign were our top priorities.
"And so given that this is the first time that we will have had ministerial posts in the entire 27 years that we've been a political party it is a huge step forward for us."
The Weekend Herald reported today the Greens' roles are thought to include climate change, conservation and a transport position.
