The driver was taken to Edinburgh Royal Infirmary for shock. A construction worker was also taken to hospital for minor injuries.
The Health & Safety executive is investigating.
The crane was working on the £4m redevelopment of a former market hall into apartments on Great Junction Street for the Port of Leith Housing Association.
A Scottish Fire & Rescue spokeswoman said that the crane was lifting a panel when the straps “snapped” and the load dropped on tot eh lorry from which it was being lifted and a car alongside it.
The incident happened at approximately 10.40am. By 1pm the road was reopened to traffic.