Apparently, the current method to auto-start Docker containers (from Docker 1.2) is to use restart policies. This will control how Docker should handle starting of the container upon startup and re-starting of the container when it exits. I've used the 'always' option so far, and can confirm that it makes Docker auto-start the container at system boot:

sudo docker run --restart=always -d myimage

Documentation Excerpt

Restart Policies Using the --restart flag on Docker run you can specify a restart policy for how a container should or should not be restarted on exit.

no - Do not restart the container when it exits.

on-failure - Restart the container only if it exits with a non zero exit status.

always - Always restart the container regardless of the exit status.

You can also specify the maximum amount of times Docker will try to restart the container when using the on-failure policy. The default is that Docker will try forever to restart the container.

$ sudo docker run --restart=always redis

This will run the redis container with a restart policy of always so that if the container exits, Docker will restart it.

$ sudo docker run --restart=on-failure:10 redis

This will run the redis container with a restart policy of on-failure and a maximum restart count of 10. If the redis container exits with a non-zero exit status more than 10 times in a row Docker will abort trying to restart the container. Providing a maximum restart limit is only valid for the on-failure policy.