A director on the board for the 10 years, Eddings would serve until next October's annual general meeting when the post would be filled by a candidate nominated after consultation with the state associations, Cricket Australia said.
Eddings takes on the role at a difficult time for the game in Australia after last year's bitter pay dispute with the players was followed by a ball-tampering scandal in South Africa in March that rocked the sport.
Peever resigned a little more than a week after being re-elected for a second three-year term, bowing to public pressure in the wake of a scathing review of the board's governance and culture.
"There's no question this year has been a challenging one for cricket and for Cricket Australia. Our aim is to ensure management are empowered to rebuild trust and strengthen the game," CA director Jacquie Hey said in a statement, "Earl's involvement over the past decade provides continuity in a time of change and enables the board to maintain strong relations with the ICC (cricket's governing body), other member countries and our valued partners." The Longstaff report, which was not released until after Peever's second term was secured, found the board partially culpable for the events in Cape Town which led to bans for Steve Smith, David Warner and Cameron Bancroft.
The report also said CA was perceived as "arrogant" and "controlling" by stakeholders and had nurtured a culture of "winning without counting the costs".
"Cricket is only what it is in Australia because of the commitment of our people, the fans and the thousands of volunteers around the country," Eddings said.
"I am honoured to serve in this position and am committed to making cricket stronger and a game that we can all be proud of."