The accident, which is described as "serious" by police happened at around 5.35am on Thursday, September 28 between junctions 34 and 33 of the southbound carriageway.
Wyresdale Road has also been closed between Grab Lane and Newlands Road.
The driver of the lorry is described as "conscious and breathing" by police but the current extent of his injuries are unknown.
A police spokesman said: "We had a report of a lHeavy Goods Vehicle coming of the motorway and going down an embankment.
"It landed near to Wyresdale Road and Newlands Road.
"These roads together with lane one of the M6 have been closed."