How to access iPhone contacts with a broken screen?

My phone screen is blank black and unresponsive. It's been like that for a few days so today I went and got an iPhone 6 because it was time for a new phone anyways. I would like to get my contacts off my old phone if possible.
I didn't back up my information to iCloud, so that's not an option. I tried plugging it in to do it via iTunes, but in order for iTunes to do that I need to authorize access to it on my phone… Which I can't do because my screen is unresponsive. Does anyone know a way around this?

Thats why you backup before you actually need it. This will be a good learning experience for you because now you get to re-enter all them again.

If you have created a backup you can access it from iTunes and I would like to recommend you the best security app that will be useful for you to secure all your media and that is Leo Privacy Guard.