Grant, who has also coached West Ham and the Ghana national team, signed a two-year contract and succeeds Aljosa Asanovic who quit over a pay dispute.
The 67-year-old's first task is to qualify Zambia for the 2023 Africa Cup of Nations having missed out on the last three editions.
Zambia, who were African champions for the only time in 2012 and are currently ranked at 88 by Fifa, are in the same qualifying group as Ivory Coast, Comoros and Lesotho.
"I'm very happy to be here. I think Zambia has potential," Grant told a news conference.
Grant took Ghana to the final of the 2015 Africa Cup of Nations where they lost to Ivory Coast in a penalty shootout.
In the 2017 edition, Ghana finished fourth.
Zambia's football federation boss Andrew Kamanga expressed confidence that "we will be back among the elite of the African game".
He said appointing Grant signals "a new dawn" for Zambian football.