A pro-regime militia chief said the fighting killed at least 40 rebels, including the leader of jihadist group Jund al-Sham, Khaled al-Mahmud, better known as Abu Suleiman al-Muhajer.
The offensive is one of two major operations by the army near the border with Lebanon aimed at cutting off rebel supply routes.
The army, backed by fighters from Lebanon's Shiite Muslim movement Hezbollah, is also fighting to seal the border in the Qalamun region, where it seized the last major rebel bastion of Yabrud on Sunday.
Charles Lister, a visiting fellow at the Brookings Doha Centre, said he expected the army to win more strategic victories along the Homs and Damascus provincial borders with Lebanon.
Losing the border would have limited effect on the opposition's weapons supplies, he said, but "in terms of general logistics, communications, and small localised safe havens, its value is well established."
As the army shelled the area around Al-Hosn, dozens of people tried to flee to Lebanon, and the Syrian Observatory for Human Rights said 60 people were "killed or injured" in the process.
A Lebanese security source said some 60 people were wounded by army fire as they crossed the river that divides the two countries.
The mainly Sunni Muslim area of north Lebanon, whose residents are largely sympathetic to the rebels fighting to overthrow Assad, has come under frequent cross-border shelling by his forces.
It is home to thousands of families who have fled the three-year-old conflict.
Smuggling routes used by opposition forces to move fighters and weapons back and forth between Lebanon and Syria pass through the area.
- UN aid convoy -
Meanwhile, an 82-truck convoy carrying vital humanitarian supplies crossed the border from Turkey, in the first such UN operation in the region since the 2011 start of the conflict, the United Nations and a Turkish official said.
The convoy is carrying food, blankets and mattresses, family kits, hygiene kits, medicines and medical supplies for "hundreds of thousands of people desperately in need in the north of Syria," said the UN regional humanitarian coordinator, Nigel Fisher.
Non-governmental organisations have previously used the border crossing to ferry aid.
In other developments, an NGO said at least 600 Kurdish residents of villages in the northeastern province of Raqa had fled after an ultimatum from the jihadist Islamic State of Iraq and the Levant.
The Syrian Observatory for Human Rights said ISIL had detained dozens of Kurds, releasing them with a warning to leave their village in the province where the jihadist group has a strong presence.
More than 500 had entered Turkey, with others fleeing to the nearby town of Ain al-Arab, it said.
Away from the front, a joint international mission said Syria has so far surrendered more than half of its chemical weapons arsenal.
The UN-Organisation for the Prohibition of Chemical Weapons (OPCW) mission said that, as of Thursday, 53.6 percent of Syria's chemical weapons material "has been removed from (Syria) or destroyed."
Syria remains behind schedule for the destruction of its entire arsenal by June 30, a deadline agreed by Russia and the United States last year.
The deadline came as part of a plan to avert US-backed military strikes in the wake of deadly chemical attacks outside Damascus last year blamed by the West on Assad's regime.
But the OPCW noted earlier this month that Syria had increased the pace of removing or destroying its arsenal after missing deadlines.