How to back up my contacts and stuff from my iPhone?

I know how to actually back it up, but the thing is, my iPhone screen is broken to the point where I can't use it anymore. Every time I connect to iTunes, it asks me to unlock my phone, which I can't do, due to the screen. I also tried iCloud, but I haven't backed up to iCloud. I've downloaded CopyTrans to see if that would work, but it also asks me to unlock my phone.

I filed a claim yesterday and my new iPhone was delivered earlier this afternoon. Now I just need to figure out a way to transfer all my contacts and apps and such.

I want to know if there's any other way I can back it up?
Please no silly answers, or answering some random nonsense just for the points, or degrading iPhones, the phones are great, its just I had a little anger issue and destroyed my phone, now I'm regretting it.

Thanks!

Added (1). poter and nicoleas, neither of those work. They both ask me to unlock my phone, which again, I can't do. >.>

You may try this phone to phone transfer. I used to use it to copy my contacts from my iPhone 4s to my iPhone 5s, it worked pretty well. Hope it helps you.

IPhone Transfer http://www.video-dvd-ripper.com/iphone-transfer/ software may be the best solution to you. It can copy contacts, photos and other files to your computer for backup easily.