Restoring Mac from backup dmg

Is there a way to restore a hard drive from a dmg back-up file not saved via Time Machine?

I have an iMac (early 2009) running OS X (10.11 - El Capitan).

I normally shutdown every night after use until I get home from work the next day and has continued to perform with no issues. However, one evening a month ago, the computer wouldn't boot past the Apple screen with a progress bar only at 3/4 and proceeded to freeze - (White Screen of Death).

When this occurred, I panicked since I hadn't backed up my data for at least a year prior. I did manage to save an image file via Disk Utility (NOT Time Machine). I then proceeded to run First Aid (no issues). I successfully restored the computer to the previous Time Machine backup from a year earlier; however, I never was able to successfully restore from my saved dmg back-up file.

I brought the Mac to Apple a week ago and they ran all sorts of diagnostics on it ... which it passed with flying colors.

With permission, he wiped the hard drive and reinstalled the OS. I have now been able to mount the image and can see all my folders/files; but I can't seem to restore them since the migration assistant doesn't recognize the file, only Time Machine files. Not sure if it makes a difference, but when we reinstalled the OS, the Rep set the computer name as "test" to get past set up to see if the computer would launch correctly.

I do plan on upgrading, but really hope to salvage all my data prior to migrating to a new Mac.

Thanks.