I had this kind of problem lately ;). It's a bit complicated, but I'll try to explain it in the simplest way.

Put somewhere in your extension (maybe options page) link "login with facebook" that redirects to: https://www.facebook.com/dialog/oauth?client_id=<APP_ID>&response_type=token&scope=<PERMISSIONS>&redirect_uri=http://www.facebook.com/connect/login_success.html

When someone clicks on this link, will be asked to give permissions, etc (facebook page). If someone would agree, the page will load: http://www.facebook.com/connect/login_success.html with parameters access_token and expires_in in address.

Now, background page should have script like this:

var successURL = 'www.facebook.com/connect/login_success.html';

function onFacebookLogin(){
  if (!localStorage.getItem('accessToken')) {
    chrome.tabs.query({}, function(tabs) { // get all tabs from every window
      for (var i = 0; i < tabs.length; i++) {
        if (tabs[i].url.indexOf(successURL) !== -1) {
          // below you get string like this: access_token=...&expires_in=...
          var params = tabs[i].url.split('#')[1];

          // in my extension I have used mootools method: parseQueryString. The following code is just an example ;)
          var accessToken = params.split('&')[0];
          accessToken = accessToken.split('=')[1];

          localStorage.setItem('accessToken', accessToken);
          chrome.tabs.remove(tabs[i].id);
        }
      }
    });
  }
}

chrome.tabs.onUpdated.addListener(onFacebookLogin);

And this is how you get access token from facebook. Now you can send requests to https://graph.facebook.com.

Hope I helped ;)

The zoom level for an iframe (which is how the Facebook Login button is shown) in Chrome is not the same as the zoom level for the page itself.

Therefore you need to click on the button and hit Ctrl + - to zoom out on just that iframe.

It can be tricky because clicking on it will trigger the popup, so you may need to use Tab to give it focus.

Note that opening Facebook in another window and zooming that to 100% also doesn't seem to be using the same zoom level - (even though the iframe appears to be the same domain).

This also only seems to be a problem with the mobile device testing tool.