First up, there's other destinations besides the US. From what I can tell, Europe, Canada, and Oceania are easier to get working Visas for than the US 
Games companies often do hire from overseas because they're always on the lookout for talent, wherever that talent is. Generally you'll do some interviews over the phone or Skype, and then sometimes they'll do a final interview in person (if this is required, they'll likely pay for your flight there and back to do this).
If they want to hire you, they'll apply for your visa for you, as they'll likely be a "sponsor" of the visa.
In some countries this isn't even required though! e.g. in some European countries you can just fly there without a visa and then visit the town hall to register yourself as a foreign worker at a skilled company, and you'll be granted permission to work.
If they're a big enough company, they'll also pay your relocation costs to move you to their country, and likely give you a temporary apartment to stay in until you find your own. If they're a smaller company, they will want you to pay for all these expenses yourself.
That said, hiring people from overseas is very expensive and much harder than hiring locals... Companies aren't likely to go through all this trouble for a fresh graduate, as they can likely get graduates from a local university without going through all this trouble.
As always, make sure you're working on games projects outside of your university work, so that you're not "just another graduate", but have something that will show off your skills.
Also, if you can get some experience at a local company first, that will be a big help (again, so that you're not "just another graduate", but have some experience).
