I visited www.mysite.com and +1'd my website.
I then visited mysite.com and the +1 wasn't there.
How can I make google plus-one treat these two URL's as the same page?
You can't.
www
is just a subdomain like any other. it is in fact different than mysite.com
and should be treated as such. You should decide which one you want to use and redirect the other one to it with your server, for example an apache rewrite rule.
You can do this very easily by specifying the canonical relationship on your pages. The canonical relationship link tells Google (and potentially other search engines) what the real URL is for a given page. This is helpful for dynamically generated pages as well as subdomains like your case.
Google+ uses the canonical link to help keep count of the +1s.
See the following topics:
Canonical links: http://support.google.com/webmasters/bin/answer.py?hl=en&answer=139066
and Google+ order of picking a target URL: https://developers.google.com/+/web/+1button/#target-url
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