How can I capture track an AdWords user from outside of the browser?

I have a PC (Windows, OSX, Linux) video game on the Steam marketplace and an extremely common problem for us developers is that there is no way to find out if a user who clicked on our game through our AdWords campaign, has actually bought it. Steam provides some analytics tracking up until our game page, but it falls into a black box after that.

I have all the power in the world once the user opens our game, so my thought here is that surely we can find the AdWords tracking ID/cookie and manually POST that to our Google Analytics platform?

I suppose step 1 is - How do we make our AdWords campaign leave a long lasting cookie/tracking ID somewhere? Step 2 - How do I find it and POST it?