The Manchester City left-back thumped a 30-yard strike into the bottom right-hand corner beyond the despairing dive of Hugo Lloris with 10 minutes to go, cancelling out Paul Pogba's first-half tap-in.
Fresh from a promising World Cup campaign in which they reached the quarter-finals, Didier Deschamps' side were in the headlines over the weekend as UEFA chief and former France international Michel Platini suggested Franck Ribery could face sanctions if he refuses to play should he be called up, despite having announced his retirement.
As hosts of Euro 2016, France men will have to make do with non-competitive fixtures to prepare for the finals in two years' time but they were given a stern test by their hosts on Sunday, as a Serbia side awash with Premier League talent continued to pose a threat throughout the full 90 minutes.
In Advocaat's first match in charge since taking the reins in July, the likes of Zoran Tosic and Dusan Tadic were particularly impressive for the hosts.
Deschamps made seven changes from the side that defeated Spain 1-0 in the Stade de France on Thursday, with Loic Remy rewarded with a starting berth after coming off the bench to net the winner in Paris.
After the Chelsea man had dragged a shot narrowly wide, Pogba put France ahead with a simple finish in the 13th minute.
Remy Cabella's corner from the left was allowed to bounce across the six-yard box, and the Juventus midfielder was on hand to tap home at the far post unchallenged.
Tosic created an opening for himself midway through the half following some delightful link-up play with Tadic, but his strike from the edge of the area was narrowly off target.
Serbia skipper Branislav Ivanovic headed Tosic's free-kick against the crossbar four minutes before the break, but the referee had blown for a foul in any event.
The hosts got stronger as the half progressed, and Tadic fired just wide after France had failed to properly clear Tosic's corner.
France were denied a second immediately after the restart when Vladimir Stojkovic reacted smartly to keep out Cabella's strike from the edge of the area.
The visitors ought to have doubled their lead in the 56th minute, but Remy sent the ball wide from short range under pressure from Ivanovic.
Stojkovic was called into action three times in quick succession shortly afterwards, the goalkeeper saving from Pogba, Remy and Cabella as France continued to press for a second.
As the game became even more open heading into the final half an hour, Kolarov twice stung the palms of Lloris with rasping drives, before the full-back made it third time lucky with a sweetly-hit shot.
Advocaat's side could have won it at the death, but Lloris tipped Ivanovic's header onto the post before clearing the loose ball with his foot.