Dempster will take over as chief executive of the relegation-threatened Scottish Premiership club on June 1, a role that the under-fire Petrie has filled on an interim basis.
He will continue as non-executive chairman and said in a statement: "We are delighted that Leeann is joining Hibernian. I am certain she will be a tremendously important signing and will provide the daily leadership needed next season to help achieve the board's plans and meet the aspirations of supporters.
"Leeann is an outstanding football leader with a proven track record. In addition to a desire for sporting success, I know from our numerous discussions that there is a shared belief in the direction of travel towards increased involvement of supporters and greater community engagement."
Dempster said: "I believe that to succeed in a sustainable way in the long term we need to grow club resources, re-engage lapsed supporters and strengthen the club collectively.
"As a big city club, Hibernian has a fantastic opportunity to build on the great physical assets and grow attendances through increased engagement and success on the pitch. I am looking forward to the challenge.
"There are a number of initiatives that we will be able to announce to supporters in due course and that will show we want to meet the shared ambitions of supporters. We want a vision that everyone can buy into so they can play their part in the future of our club."