The former Motherwell chief has taken over from under-fire chairman Rod Petrie, who is still facing supporter demands for him to quit.
Angry fans, led by former midfielder Paul Kane, have launched the 'Petrie Out' campaign after their side suffered a humiliating season which began with a record Europa League defeat to Malmo and ended with them being relegated from the Scottish Premiership after a play-off final defeat to Hamilton.
But Petrie hopes the appointment of Dempster will sooth simmering tensions in Leith.
Dempster will now be in charge of "all club activity", including the club's football department, supporter engagement, marketing and press policies and, crucially, Hibs' books.
Petrie has given up all of his executive responsibilities and has now taken up the role as non-executive chairman - but will remain as the club's main link to major shareholder Sir Tom Farmer.
He said: "The Board and I have recognised for some time that change was needed at the Club, and we set out to recruit the best possible person to take the Club forward. Leeann is that person, and we are delighted that she officially starts work with us today.
"Supporters could not have been surprised if Leeann had second thoughts following the enormously disappointing end to our season which saw us relegated to the Championship. But we know the character of the lady and she never wavered in her determination to join Hibernian."
Dempster's five-year stint in charge at Fir Park saw the Steelmen notch up three consecutive top-three finishes in the top flight.
Now she will be charged with producing similarly impressive results in Edinburgh.
Dempster said: "There is a lot to be done at Hibernian to return the club to its rightful position in Scottish football - both in terms of success on the pitch and in the way we engage with our supporters and our community.
"I am excited by the challenge, and I will be relentless in pursuit of improvement and success. However, we need everyone to be together, to work together, and to ensure that we maintain the stable platform at the club which is vital to help propel us towards our common goal of promotion back to the Premiership next season.
"I understand people are upset and concerned about relegation, but they should understand that I am confident that I have the authority and freedom to make the decisions I want. I am also very confident that with everyone working together we can improve things quickly at the club."