How to back up iphone 5 with broken screen?

I broke the screen on my iPhone 5 and the digitalizer doesn't work, my phone has a lock on it (which i can't unlock as the screen doesn't work) so I can't sync it by iTunes to create a back up. The phone works and switches on but I can't see the screen either. Is there a program I can use to back up my phone as I will be receiving a replacement.

There are several programs such as iExplorer and iFunbox that you can download in order to accomplish what you want to do. They will allow you to manually browse your phone and save whatever data you need. You may be limited at the amount you can back up until you purchase the full version rather than the demo, which I'm assuming is worth it if your data is important enough. It's either that, or the second option you have is having a cell phone professional nearby replace your screen and you will be back in business. Although replacing the screen might cost you more, it might be the safer bet.