to keep all critters from going under your fence either buy some cheap wire fence or craig's list search for some free used chainlink fence.
You do not have to bury it. Laying it flat is good enough. in weeks it will be beneath the grass. in a year it will be beneath the soil.
If possible slide it under the fence and around the fence posts a bit so that it stops animals from going in or out.
note: If you have a dog that is digging out under the fence chicken wire might do the job. It will only last a year or two in the ground, but most dogs give up and don't try after being unsuccessful.
"Those aren't poodles. They're Dobermans with afros."