Pull out individual golden carpet sedum plants manually by clutching the base as close to the soil as possible and pulling slowly up to remove the root along with it. If the root isn't removed, the plant will quickly grow back. Dig out the root with a hand shovel if the stem breaks off without the root coming out with it.
Cover large areas of golden carpet sedum with a plastic tarp then cover the tarp with 2 inches of mulch. While the plant grows in various conditions, this blocks the sun entirely and also prevents any rain exposure, killing the plant off over time. This is less labor-intensive but takes longer.
Spray any remaining golden carpet sedum with an herbicide to kill it off. Follow the packaging instructions to ensure that surrounding plants aren't affected and the herbicide works as required.