The proposal was voted for unanimously by New Zealand's 26 provincial unions and the New Zealand Maori Rugby Board (NZMRB), although it still needs sign off from the New Zealand Rugby Players' Association (NZRPA).
The private equity deal would see Silver Lake inject 387.5million New Zealand dollars into NZR, coming after the governing body recorded an operating loss of18.7m NZD at its annual general meeting in Wellington.
The controversial deal has been mooted for some time and takes the game a step closer to three-time world champions the All Blacks becoming a non-wholly publicly-owned entity for the first time.
"We are thrilled that our Provincial Unions [and] the Maori Rugby Board have recognised the importance of private equity in driving commercial revenue and enabling investment to ensure rugby thrives and survives into the future," NZR chairman Brent Impey said.
"Today's vote for Silver Lake represents a transformational opportunity for our game and one we must grasp."
Ongoing discussions with the NZRPA will be critical to establishing the terms of the deal, although Impey was disappointed they had not yet given their consent.
"Through swift action, good governance and hard work of our people we are fortunate to be one of the best placed national unions in the world," he said.
"However, we are at a critical juncture and need our players’ support if we are to make the most of the opportunity in front of us.
"The game has to change, and Silver Lake's capital injection would allow us to re-imagine rugby and invest in the areas of the community game that need it most, particularly teenage and women's rugby, and to create better and more engaging experiences for our fans.
"We hope the NZRPA will realise the significance of the opportunity in front of us and will continue to work toward an agreement in coming weeks."
New Zealand's sports minister and deputy prime minister Grant Robertson also weighed in on the matter, calling for the NZRPA to come to the table for talks.
"New Zealand Rugby has been working hard on it, trying to establish what it believes is a more secure financial base for the sport," Robertson said.
"Some of the details that are now emerging show the provincial unions will be getting some slice of the extra money that might be coming in, but other details are still a little bit unclear.
"Obviously on the other side of the equation you've got the Rugby Players' Association who are concerned about both the salaries that the players get but also some other issues like protection of cultural icons such as the haka."