The deal paves the way for a June start on the new stand with the previous design team being automatically re-engaged to speed up construction plans to enable the project to be completed in time for the 2019 Cricket World Cup.
The council found the financial services investor after a previous scheme to pay for the stand redevelopment fell through.
As part of the deal, the council will enter into agreement with the investor to lease the new facilities.
Leeds Rhinos has also pledged £5m towards the total cost of the redevelopment.
Leader of Leeds City Council Cllr Judith Blake, said: “I am pleased that executive board has backed the deal which we have arranged with a private third-party investor to fund the redevelopment of Headingley Carnegie Stadium.
“With the tight timescales which are involved for this work to begin, this route offers the best and most viable option moving forward. As part of this agreement, the cost of financing the development will be met by both the rugby and cricket clubs at no cost to the tax-payer.”
The Leeds site is home to rugby league side Leeds Rhinos, rugby union side Yorkshire Carnegie and Yorkshire County Cricket Club.
Under the £40m redevelopment proposals a new rugby south stand and a replacement for the existing joint stand, which overlooks both the rugby pitch and cricket ground would be built.