How do I backup my shattered iPhone that has a passcode? So over the Halloween weekend.. I shattered the screen to my 5S :(

My buddy is lending me their old iPhone 5 for a bit. I planned to backup my 5s and restore it on the 5. Problem is.. I keep getting the message in iTunes that says I have to enter my passcode on my 5s to start the backup!

The screen is absolutely destroyed. It doesn't recognize fingers or anything. The screen only turns on when I get a notification.

I was thinking it might be possible to replace my 5s screen with my buddy's 5 screen if all else fails, but I'm not sure if thats possible. Maybe someone can answer that too.

Thanks guys!
If you phone is on, if you plug it into your Mac or computer it should automatically start to backup. As long as you have synced it on that computer at some point. Then you can just restore it when you get a new phone.