It happened during the middle of the night on the A55 in North Wales between Llanfairfechan and Conwy tunnel in Gwynedd.
Last night North Wales police defended their actions.
But on Facebook there was outrage from animal lovers who questioned why the main route wasn”t closed and a dog handler called in.
One woman wrote: “The A55 at 3am is hardly busy is it and we”re all used to hold ups on it anyway. Totally outrageous action.”
Another wrote: “If a mentally unstable person were running around frantically, they would have run them over too?!
“The police have specially-trained dog handlers in the force, they could have slowed the traffic down and got a dog handler to catch it. I”m outraged.”
The death of the dog, believed to have been a foxhound, follows up from police tasering a sheep on the same road.
Chief Inspector Darren Wareing said: “Shortly after 3am on Monday several calls were received in the police control room concerning a dog which was running loose on the A55. Officers from the Roads Policing Unit attended.
“Despite several attempts to catch the dog, it continued to run in and out of traffic. At one point an officer tried to take hold of the dog but was then bitten. Further attempts were made to catch the dog, which was by now running in the middle of an unlit carriageway with approaching traffic having to take avoiding action at speeds in excess of 70mph.
“One car and a HGV had to swerve to avoid the dog whilst the officers remained on foot in the carriageway.”
He added: “The potential for a serious collision was present throughout. In the circumstances there was no alternative way that officers could contain the dog and minimise risks to motorists.
“The only safe option was to run the dog over at sufficient speed to ensure that it was destroyed and would not suffer. Other methods of destruction were considered, but were ruled out on the grounds of public safety.
“Fast roads such as the A55 present inherent risks and to have vehicles including an HGV having to swerve in the dark was deemed unacceptable as the officers witnessed a number of near misses, and were highly concerned about the potential aftermath of a high-speed collision.
“Both officers have their own dogs and did not take this decision lightly. Due to the seriousness of the incident it needed bringing to a conclusion quickly for the safety of all concerned.”