The fighting erupted yesterday in an undemarcated area of the frontier between Qalamun in Syria and Arsal in Lebanon, the Lebanese official told AFP, speaking on condition of anonymity.
"A Hezbollah fighter was killed and 12 were wounded on Saturday night," he said. "The fighting intensified on Sunday evening with three Syrian fighters killed and 10 wounded."
The long and porous border is often used by smugglers, refugees and fighters.
Syrian Observatory for Human Rights head Rami Abdel Rahman said: "It appears Hezbollah launched the attack in a bid to finish off the pockets of rebel resistance."
In April, Syrian forces backed by allied fighters from Hezbollah retook control of most of the Qalamun region.
But Syrian activists say hundreds of opposition fighters have taken refuge in the caves and hills in the border area, using it as a rear base from which to launch attacks inside Syria.
Last month, Lebanese army forces carried out raids in the area targeting militants with ties to "terrorist groups", an army statement said at the time.
