Don't dig down.
Rather than digging down and having the wire simply being an underground extension of the fence you can make an "apron". You can do this by digging out the top couple of inches of soil going outward from the outside of the fence for 12" to 24". Bend the fence wire outward at a 90-degree angle and press down against the fresh dug ground. You can make some homemade "staples" to hold the wire down snug by clipping some coathangers in to two or three pieces and bending them into a "U" shape. When you have the wire in place and secured then pull the soil back over it. Grass will easily grow through the wire but when animals walk up to it and start digging they will hit the wire...they're not smart enough to back off and try digging at the wire's edge. If the wire is buried well you can even cut grass over it. Some people don't bury the wire but simply have it laying on top of the grass and held down by bricks, rocks, whatever,...the grass will still grow through it but I wouldn't really want to run a lawnmower over it.
You can do this a couple of ways... One way is by bending the bottom of the fence at a 90-degree angle as you install it. Another way is to install the fence as you normally and then come back and add a piece of fence bent in a 90-degree angle to the main fence by either attaching it with wire or using hogrings to secure the short apron "L" to the fence.
I hope this helps.
Best wishes,
Ed
