Sure, as long as you’re authenticating correctly. You’ll need to use LOGIN or PLAIN auth. Also, make sure you’re using the right machine as a smarthost - you wouldn’t want to use “example.com” - use “mail.example.com”, and make sure Exim isn’t looking up the MX for that, but is sending to the A records for mail.example.com (in Postfix, you have to put brackets around the hostname or something to do this).
Unfortunately, while I can promise this will work if you set it up right, this isn’t something we can provide support for.
Have you looked in your mail logs yet? The answers are probably there.