Very cool that this worked, so I figured I'd share online in case it happens to you. The screen went bad on my iPhone 5, so I couldn't tell the iPhone to trust iTunes in order to make a backup. It kept prompting me to enter a passcode which I couldn't enter.
Hold your power and home buttons down for 10 seconds. The phone will restart. Keep holding both buttons until the phone comes back on. Release the power button but keep holding home for a while. Release the home button.
Using CopyTrans Shelbee, start your backup. I got the idea from a Dr. Fone instructions, but it worked for the free backup edition of CopyTrans Shelbee