Roundup is what I use. Use it when the wind is not blowing. Any plant it lands on dies but once it hits the soil it is neutralized, it only work through absorption through the leaves. You can spray it on the plant or if you don't want a dead spot around it from over spray, you can brush it on the leaves with an artist" or small paint brush. It is best to put it on when the plant is growing and about 6" tall. Don't be in a hurry it will take about 7 days before it looks like it is dying but it will kill it all the way down to and including the roots.
One very wonderful thing about thistle is the the seeds can lay in the ground for many years before they germinate, so you will probably have them for a while.
Be sure you live your life, because you are a long time dead.-Scottish Proverb