TV4 news found the video in which a rebel fighter emerges with a Carl Gustaf rocket launcher perched on his right shoulder.
The report dubs the weapon a "Swedish export success story".
It is the second time in the past few months that Swedish weapons have made their way to countries to which Sweden does not export arms.
The last case was unearthed in Burma, to which the European Union forbids all weapons exports.
Pictures taken by a Burmese freelance photographer and published in the Swedish media in December 2012 showed a Carl Gustaf M3 anti-tank rifle and ammunition left behind by Burmese government soldiers. They were recovered by Kachin rebels after recent clashes.
A serial number was traced back to a shipment from Sweden to India, where the government promised to investigate the matter.
As the new example of stray weapons emerged on Wednesday night from Syria, a Swedish military expert confirmed that TV4 had correctly identified the "effective, popular and easy-to-use" Carl Gustaf model.
"Sweden has sold this model to about 40 countries," said Christoffer Burnett-Cargill at the Swedish Peace and Arbitration Society (Svenska freds).
"It shows our export checks and balances don't work. Yet again, Swedish weapons have been diverted."