The white Fiat Punto flipped onto its roof near Junction 12 southbound at Uddingston at about 15:30 on Saturday.
A man and a woman were freed from the car and taken to Wishaw General Hospital.
No other vehicles were directly involved in the crash, but two other cars were damaged by flying debris.
The motorway was closed in both directions for a time but the northbound carriageway later reopened.
The southbound carriageway remained closed with diversions in place.
Two people have been taken to hospital after their car overturned on the M74 just outside of Glasgow.