The incident happened on the A6 just north of Buxton at 2pm on Friday.
A spokesperson for Derbyshire Roads Policing Unit said: “Through no fault of their own, the occupants of the car had a near death experience when a lorry blew over on top of them on the A6 at Dove Holes.
“Amazingly there were only minor injuries and their dog was OK too.
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"Time to heed the warnings about Storm Eunice and stay at home,” they added.
The A6 had to be closed to allow specialist services to recover the lorry.
Following the incident, police urged drivers of high-sided vehicles to avoid exposed routes in Derbyshire – in particular across the Peak District.
They added: “With the current issues being caused by the wind and rain, unless you really need to travel then simply don't head out.”
Storm Eunice is causing disruption across Derbyshire and the rest of the UK.
A Met Office amber wind warning ends at 9pm on Friday.
Members of the public are being urged to take notice of this vital public safety information.