If you think that employers are doing this because an H1B worker is cheap, you're very badly deluding yourself. Lets take a look at a case study I know well (myself). To hire a US worker to do my job (assuming one existed, which the lack of our ability to hire a second me suggests they don't just now) it would have cost them approximately $140k a year, plus some sign on bonus in the region of $100k of stock and $10k cash. They might have had to pay around $10k in moving expenses if they'd lived on the other side of the country. On the other hand, to hire me, it cost them in the ballpark of $170k a year (because I used the fact that my wife would not be able to work as leverage to get a higher salary, plus they have to pay over the prevailing wage anyway), closer to $150k of stock on sign on, and in the ballpark of $20k in cash (because I used the enormous risk of moving to another continent to justify that my sign on bonus should be larger). It cost them in the ballpark of $30k to ship my stuff across the Atlantic. It cost them $15k in plane tickets *just* to interview me (typically an 8 hour or longer flight results in it being business class), and a further $30k to bring be over for initial training before my H1B began, and then to fly both my wife and me out here. It cost them thousands in paying for accommodation and a relocation agent to get my started finding somewhere to live because I couldn't do that from the other side of the Atlantic, and because I needed assistance getting started with credit due to having no US history. It cost them thousands just to get the visa in the first place, and it costs them thousands to work towards getting me a green card.
Long story short. Over 2 years (a typical length of stay at a tech company in the bay area) I cost my company substantially more than a US citizen, due to the extremely high up front costs, and the slightly higher rolling costs to them.
Frankly, the idea that an H1B worker is "cheap labour" and used to suppress wages and get a cheap way out is so ridiculous it's beyond belief.