The northbound articulated lorry hit the barrier on the A74(M) south of Lockerbie, at junction 20, and left the road just after 5.45am on Monday.
Both carriageways were closed for efforts to recover the vehicle, with a crash scene investigation taking place.
The southbound carriageway reopened around midday but the northbound carriageway remained shut.
Inspector Campbell Moffat said: "We are appealing to anyone who may have been on the A74(M) motorway this morning to get in touch if they saw this crash.
"We would also be keen to hear from anyone who may have captured any dashcan footage of the crash, or the northbound lorry just prior to the crash.
"Anyone with any information should call police on the 101 number and quote the reference number 0454 25 November 2019."