The takeover has been approved by the Scottish Football Association, Rangers said in a statement, adding that the new leadership has committed to invest 20 million pounds ($26.95 million) into the club.
"This new ownership phase represents a significant step forward for Rangers FC," said Rangers chairman Fraser Thornton.
"The incoming shareholders bring not only funding but also deep expertise in strategic planning, infrastructure development, and sporting excellence."
Rangers finished their domestic campaign second behind champions Celtic for the fourth consecutive year, last winning the league in 2020-21. Both teams are tied on 55 championships.
"This club's history and traditions speak for themselves, but history doesn't win matches," health insurance tycoon Cavenagh said.
"We know that the true way to honour the club's heritage will be to drive performance. Our focus is simple: elevate performance, deliver results, and bring Rangers back to where it belongs - at the top."
Rangers will compete in the Champions League second qualifying round for a chance to play in next season's competition.