The couple filmed the young bears running ahead of them on a winding road between La Lastra and Tudancia in the northern region and posted the video on social media where it went viral.
But the conservation group Fundacion Oso Pardo (FOP) - Brown Bear Foundation – which works to protect the endangered species has condemned the footage as evidence of “animal abuse”.
Instead of slowing their vehicle when they came across the bears, and allowing them to calmly make their way off the road, the car drove as close to the animals as possible with the headlamps on full beam in order to film the encounter.
One person in the car can be heard whistling to attract the attention of the bears while the other comments: “Look at them running together!”
When one bear is chased off to the side of the road, the car continues to speed behind the other as it continues its panicked run. The video lasts almost a minute.
Writing on their Facebook page, FOP said the car chase was “iIlegal because abusing endangered animals is prohibited” and “dangerous because it could cause a traffic accident with oncoming cars.”
It advised that the correct behaviour is "to slow down and allow the bears to leave the road"
The conservation group said the latest video was part of a worrying rise in such cruelty as people sought to capture encounters on their smart phones.
There are an estimated 280 brown bears in Spain, mainly in forest-rich Asturias and Cantabrian mountains and the Pyrenees on the French border.
But the naturally shy animals can antagonise locals as they venture out of the woods and raid fruit orchards and bee hives in search of food when drought causes their natural food sources to become scarce.
Farmers also complain about the impact the bears have on their livestock. Earlier this year, a flock of 200 sheep were found dead at the base of cliff in Catalonia where it is believed they had been chased by a bear.
Male bears weigh as much as 350 kilos (770 pounds) and females 200 kilos, and they can easily outrun a human. Rearing up, they stand up seven feet (two metres) tall.
If the bears are not sufficiently fattened up before their winter hibernation they may not survive.