The road was shut in both directions for several hours between the A6136 (Catterick) and the A66 (Scotch Corner) shortly before 9am.
Police say no-one was injured in the incident.
The northbound and southbound carriageways have been reopened but traffic is still expected to be slow.
The truck, which seems to have been carrying hay bales, is said to have burned with such intensity that it melted the road surface.
It has now been towed away and firefighters have been dealing with the burning straw.
A North Yorkshire Police spokesman said: "We've moved the lorry to a safe location, so that North Yorkshire Fire & Rescue Service can continue to tackle the burning straw.
"Residents may notice smoke in the Middleton Tyas and Scotch Corner services area. This is nothing to worry about, but please drive carefully.
"All closures have now been removed from the A1 northbound and southbound, but please allow extra time for your journey, as traffic will take some time to ease. Thanks for your patience."