It happened on Wednesday, June 19 at 3am on the M62 between junction 28 (Tingley) and junction 29 (Lofthouse).
The crash involved a car and the crane.
One person was trapped in the crane.
Fire crews from Hunslet and Cleckheaton attended and freed the person.
The injured person was taken to the hospital.
An incident statement from West Yorkshire Fire & Rescue said: "Road traffic collision involving a car and a crane. One person trapped in crane, extrication carried out by fire service personnel and transported to hospital attended.
Two appliances from Hunslet attended along with the technical rescue unit from Cleckheaton."
Vehicle extrication is the process of removing a vehicle from around a person who has been involved in a motor vehicle collision.
This occurs when conventional means of exit are impossible or inadvisable.