Eaton’s assistant Matt Bramald is Hunslet’s new team boss and will be in charge for Sunday’s pre-season game at Featherstone Rovers.
Eaton will continue as an assistant-coach at Leeds Rhinos and is set to help Hawks on a voluntary basis.
The change comes as Hawks cut costs ahead of the new season in Kingstone Press League One.
Both parties have stressed it was a mutual decision and has been made in the best interests of the club, who were relegated last year.
Eaton returned from Leeds Rhinos’ training camp in Florida on Wednesday and broke the news to his players at training on Thursday evening.
“All parties are in agreement,” Eaton told The Yorkshire Evening Post. “We think going forward it is hopefully going to stand the club in good stead.”
Eaton admitted he was “emotional” at stepping down from the post he has held since the start of the 2012 season.
He added: “I have a great affinity with the club.
“There’s some great people working here and a really strong bond between staff and players.
“I will be a voluntary consultant and I will help out in any way I can, whether that’s helping with coaching if needed or assisting the club off the field.
“I won’t be far away. Hopefully the club can continue to progress and stabilise itself on and off the field.
“It has earned respect as a well-run rugby league club and that is something I am proud of.”
Eaton has given his backing to Bramald, who played for clubs including Dewsbury, Featherstone and Hunslet and had a spell on Castleford Tigers’ coaching staff before joining Hawks for the 2015 campaign.
“I know him really well,” Eaton said.
“A decent chunk of my playing career was spent playing with him and we became close friends.
“I worked with him at Keighley and had no hesitation bringing him here as an assistant.
“He has had two spells under Daryl Powell and Ryan Sheridan at Featherstone and Castleford and he has certainly put the hours in.
“He has enough experience to warrant moving up and that’s another reason why I won’t be far away.
“Whenever he needs any help or assistance I will be there for him. He has earned his stripes and deserves a chance with his own team.”
Hawks suffered a significant cut in central funding following relegation from the Championship last season and Hunslet’s newly-appointed chief executive Colin Smith said: “It is cost-saving. The partnership will continue and Barry is going to help out with one thing and another.
“It makes sense financially. With the financial constraints on us we have got to do what is right for the club, to make sure the club is here and alive and kicking over the next so many years.”
Smith added: “I want to give Barry recognition for his part in this. He could have seen his contract out, but it is an agreement between all parties for the best for the club.”