The Real Madrid player took the captain's armband as regular national skipper Hugo Lloris sat out Tuesday's match at the Stade Velodrome.
Didier Deschamps' team dominated for the majority of the contest, but looked set to be frustrated before Varane headed home Antoine Griezmann's 84th-minute corner.
France have lost just once in 2014 - their FIFA World Cup exit at the hands of winners Germany standing as the only defeat in the calendar year - and Varane is happy to see similar standards developing to the ones he enjoys at club level.
"I am taking advantage of this moment to the maximum," he told reporters.
"To be captain, it is the cherry on top of the cake, but what matters is the team.
"The aim is to win continuously with Real Madrid.
"I am very happy to finish the calendar year on a good note."
Varane's Real team-mate Karim Benzema had an opportunity to give France late breathing space, but sent a penalty sailing over the bar after Oscar Wendt's handball.
"On the penalty, I wanted to hit the ball hard. I am disappointed but what is important is the win," Benzema told reporters.
"The better we are on the pitch, the more that the fans will support us.
"I would have liked to score this penalty. I need to work on them in training."
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