Police had warned drivers to avoid the area as vehicles were becoming stuck, but traffic is now moving in both directions.
A Sunderland City Council spokesman said: "Traffic is now moving again in both directions on the A690 Houghton Cut following the earlier closure of the eastbound carriageway.
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"Gritters have been out all night and are continuing to treat priority routes."
Durham Constabulary urged drivers to take care in the poor weather.
A force spokesman said: “Snow is causing problems on much of the county’s roads this morning.
“We’d urge people to take extra care when out on the roads.
"Please drive to the conditions, leave plenty of space and give yourself extra time to reach your destination.”
North East Traffic Live has confirmed that the damaged car from the collision has now been recovered.
The Met Office issued a yellow weather warning for snow and ice for the region until 12pm.
The rest of the day will remain cold with scattered showers and sunny intervals.
Showers will be sometimes heavy, falling mainly as snow however, showers will sometimes fall as sleet and perhaps rain, especially near the coast.
The maximum temperature will be 3 °C.
Tonight there will be large amounts of cloud and further scattered showers, mainly of snow, but perhaps sleet or rain near the coast.
There will also be a widespread frost and risk of icy stretches on untreated surfaces, with temperatures dropping to as low as -3 °C.