In Atlanta, Fulton County Board of Elections spokesman Mark Henderson said unofficial results from Tuesday's vote show the measure passed with 81 percent support.
Results tallied by The Atlanta Journal-Constitution showed voters backing the change in many of the nearly 100 locations where it was on the ballot. At least one town, Forest Park in suburban Atlanta, rejected the idea.
The issue was not on the ballot statewide. The state Legislature this year ended recurring debate by giving local communities the power to decide for themselves.
Nationally, only Indiana and Connecticut now prohibit Sunday package alcohol sales statewide.