STORY: Earlier, more than 200 people had gathered around the building in the center of the Druze-majority city, chanting slogans calling for the overthrow of Syrian President Bashar Assad, they said, amid spiralling prices and economic hardship.
"Our message today is to the regime: You see how people are dying from starvation. People are waiting for aid," one protester was heard saying in a video released a local website Suwayda 24.
Syrian state media said tens of "outlaws" stormed the governor's office and burned files and official papers.
Three witnesses told Reuters the governor was not in the building which was vacated before protesters stormed the offices.
Witnesses also told Reuters that once inside the building, demonstrators brought down pictures of Assad.
A civic activist and editor of Suwayda 24, a local website that covers the southern region, told Reuters several people were wounded in the exchange of gunshots.
A source in the city hospital said one civilian who was being treated had died from gunshot wounds while another was still in hospital after being shot.
Sweida province has been spared the violence seen in other parts of Syria since the start of the over-decade long conflict that began after pro-democracy protests erupted against Assad's family ruler were violently crushed by security forces.