Officials said crews were working to clear the route near the Salang Tunnel for ambulances, bulldozers and other road-clearing equipment.
Local people are helping Afghan soldiers dig through the snow to vehicles buried or stuck.
"There are many other cars swept away," Gen Mohammad Rajab, the head of the Kabul-Salang highway, told Reuters news agency.
"The death toll may rise as we dig out dozens of other frozen bodies."
Reporters said they could see a number of vehicles, including passenger buses, that had been swept deep into a gorge at the side of the road.
The area is often affected by heavy snow and has been hit by avalanches in the past, the BBC's Martin Patience says from Kabul.
On Tuesday, Afghan interior minister Mohammad Hanif Atmar fended off questions as to why the road had been open in the first place, insisting the situation had appeared manageable until snowstorms unexpectedly struck.
The road through the Salang Pass is the only major route over the Hindu Kush mountains linking southern Afghanistan to the north and Central Asia that remains open throughout the year.
Reaching 3,400m (11,000 ft) at the pass, the road is one of the highest in the world. It was finished in the 1960s with Soviet help.
Meanwhile, an Indian soldier was killed but 13 others were rescued after a second avalanche in two days in Kashmir.
The snow struck an army post on Tuesday in Indian-administered Kashmir, along the Line of Control adjoining Pakistani-administered Kashmir, officials said.
At least 17 Indian soldiers were killed on Monday when an avalanche hit a military training camp in Indian-administered Kashmir, the army said.
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