Advocaat left his role earlier this month following a 3-1 defeat to Denmark in Euro 2016 qualifying.
Serbia are second-bottom of Group I with only one point, having been awarded a 3-0 win following their abandoned fixture against Albania only to also be docked three points by UEFA.
Curcic will now take on the challenge of bringing fresh impetus to their qualification campaign.
A statement from the Serbian Football Federation on Tuesday confirmed Curcic's appointment.
"At today's meeting, the new coach of A team of Serbia was unanimously elected [as] Radovan Curcic, former coach of the young Serbian national team."
The new coach will hold his first press conference on Wednesday.