DAMASCUS—Syrian government forces, backed by members of Lebanese militant group Hezbollah, launched a sweeping operation Sunday to capture a rebel stronghold near the Lebanese border, according to Syrian state media and activists opposed to the regime.

Taking the town of Qusayr, southwest of the city of Homs, would bolster recent gains by regime forces in central Syria and around the capital, Damascus. It also could further embolden Syrian President Bashar al-Assad, who told an Argentine newspaper over the weekend that his...