Sussex Roads Police said officers 'followed a scrape mark left on the road' last Thursday night (April 21).
The driver travelled between Falmer and Shoreham on the A27 missing a wheel, police said.
A spokesperson added: "His trail came to an end when he had a sleep in the live lane of a dual carriageway. In custody, they blew 2.5 times over the limit."
Police said the driver was charged to appear in court for driving whilst over the prescribed limit of alcohol and dangerous driving.
National Highways were made aware of the damage and 'passed details for the costs to be recovered', in order to repair to the carriageway.