Highways England said the motorway was closed anticlockwise at junction 2, and that the A2 was shut in both directions at the Darenth interchange after "significant damage" was caused to the barrier.
Two of the three lanes of the clockwise M25 near the scene are also closed.
Six miles of congestion have built up near the Dartford crossing in Kent.
Pictures of the incident appear to show some of the lorry's wheels partially over the edge of the fly-over.
Kent Highways said emergency services were at the scene.
Delays were 45 minutes above normal journey times, according to a Highways England tweet at 2.13pm.
Drivers are being diverted around the accident.
There is so far no information on whether anyone has been injured or how the incident happened.