(Not that it's not a great program. You can do amazing thing with exim. However, it falls in to the category of bringing a bazooka to a knife fight.)
I use a package called nullmailer to forward system messages to my smtp server for delivery. It provides the standard sendmail interface to local programs, and it's only function is to forward the message to a more capable email server.
The setup steps are roughly this:
install:
sudo apt-get install nullmailer
edit /etc/mailname to hold your "from" domain name. (If you were google, you'd use "
gmail.com".
edit /etc/nullmailer/adminaddr to contain the address you want mail to root to be routed to.
edit /etc/nullmailer/remotes to contain the mail server to forward email to.
That's about it for configuration. It can also do things like authentication with the remote server if you need that.