We had this problem at our old place. We got some bread crates (the big plastic shallow version of milk crates), lined them with newspaper, filled them with soil and grew groundcovers in them until they were WELL established. then transplanted the entire sod piece into the hole left by the erosion. Because the top layer was by then held together with roots, it worked pretty well. Hope that helps.

The tubes are called fiber wattles or coir wattles or rolls. They are available online and I'm sure they have local distributors in your area.

Rolanka is the most heavily advertised.

Remember that erosion is a combination of speed and volume of water. Anything that you can do to reduce either or both of these helps. Do what you can to avoid adding water to the driveway.