County had entrusted Eriksson with the task of finding a new manager, and he has turned to his former assistant at both Manchester City and the Mexico national team.
A brief statement on the club's official website read: "Notts County are this afternoon pleased to announce the appointment of Hans Backe as the club's new first-team manager.
"Hans has signed a three-year contract and starts his duties with immediate effect."
Backe held talks with the board at Meadow Lane on Tuesday morning and replaces Ian McParland.
Like Eriksson, Backe was on the shortlist to become the next head coach of the Sweden national team, but he insists he is keen to get back into club management.
Backe has spent much of his managerial career in Scandinavia and has won four Danish titles with FC Copenhagen and Aalborg.
He has also had spells at Austrian side Salzburg and Greek giants Panathinaikos.
County's last two matches have been overseen by joint caretaker team Dave Kevan and Michael Johnson, but Backe will be in charge for the home game against Shrewsbury at the weekend.