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I can't comment specifically on your question--I went into graduate school in EE (from Engineering Physics, but basically EE), but at the same school I did my undergrad at. My school might not be the most prestigious institution in the country (as loaded of a description as that is) but I like to think it's a pretty good school (and according to some publication with whatever arbitrary criteria they use to quantify schools--graduate and undergraduate combined--we rank #127 in the world).
But in general, grad school departments (and individual advisors) will receive far more applicants (with varying degrees of seriousness / spamming) than available positions, so by no means is it simply a matter of submitting your application (not that I'm saying you've implied this).
Now, I'm given to understand that my application experience applied throughout most programs in most of Canada (and hopefully, you don't mind my use of sports metaphors). You don't apply to get into grad school and join the grad student "free agent pool" looking to get signed up by an advisor, but rather, get "sponsored" figuratively and literally (assuming they're paying you a stipend out of their research grants). That's not to say that you're locked in, or that they "own your rights", as I've seen grad students / advisors split after going on divergent research paths, hitting dead ends in their research, or just plain falling out.
Then again, my friend in Computer Science (at the same school) basically *did* have to go through the "free agent process"--your mileage varies greatly, depending on where, when, and probably how you apply.
Back to the main topic, if you have the minimum qualifications (and sometimes, even if not, if the prof in question really, really, really likes you) you have to convince individual advisors to take you on as a student based on your educational background, your familiarity with the field, your willingness to suck it up, work ethic, being the right guy/gal at the right time, etc. And then, you're usually in (sometimes the Departments will raise issues, e.g. with questionable qualifications, poor TOEFL, poor marks, etc.)
I should advise you that most places will charge international students differential tuition (usually, 100% on top of the baseline). As an American, you would be international (but thankfully, the baseline isn't all that high, comparatively). Usually, you get paid enough of a stipend, you can usually cover tuition, rent, and cost of living (assuming you don't go too extravagant on the latter two).
TL;DR version: no, it's not a cakewalk (why would you want to go to a place that accepted just anybody, regardless of whether or not they're qualified or good) but if you have good grades, letters of recommendation, and above all, interest and some demonstrable potential, you'll probably find a position. Then again, this generic advice probably applies almost anywhere and with everything.
