Wear gloves and cover as much of your skin as possible in case plant sap gets onto your skin during the removal process. The sap is a severe skin irritant that causes immediate burning and blisters. There are also sharp spines on the leaves.
Explore the ground around the agave to locate the roots. A large agave can have a large root system so start a few feet out from the plant to see how far the roots spread.
Dig the entire plant and as much of the root ball out of the ground as possible to discourage any more offshoots. Don't dispose of the roots or the leaves in the compost because the roots can send up pups and the spines will stay forever even if the leaves decompose. Send it to the landfill in plastic bags.
Slice across any roots left in the ground you haven't been able to remove with a sharp knife and apply a glyphosate herbicide with a sponge brush within 5 minutes of making the cut. If any roots are left in the ground alive they can eventually send up new agave pups. Don't spray and try not to get a lot of herbicide in your soil or you might kill whatever you plant in that spot next.
Fill the hole with new soil to prepare the area for new plantings.