Google provides API Client library:
<script src="https://apis.google.com/js/api.js" type="text/javascript"></script>
It can do google API requests for you, given the API path and parameters:
var restRequest = gapi.client.request({
'path': 'https://people.googleapis.com/v1/people/me/connections',
'params': {'sortOrder': 'LAST_NAME_ASCENDING'}
});
Since the library is served from google domain, it can safely call google API's without CORS issues.
Google docs on how to use CORS.Google docs on how to use CORS.