Police discovered a black 4 x 4 Mercedes had driven through the site fence and fallen down a 25 ft drop into a building site on Miller Street at the junction of Corporation Street.
The driver, aged 27 and two passengers, both aged 26 years, were taken to hospital for treatment.
The site was not active at the time and no-on was injured from the construction team.
A 27-year-old man has been arrested on suspicion of dangerous driving and driving while under the influence of alcohol or drugs. He remains in police custody.
Sergeant Mark Thomas said, “The driver and passengers involved in this incident appear to have had a lucky escape.
“I would ask anyone who may have either witnessed the collision or who saw the Mercedes being driven prior to the collision to contact us.
“This once again highlights the dangers of driving irresponsibly and after consuming alcohol.”