How to unlock/backup broken iPhone 5S in iTunes?

I had waterdamage on my iPhone, and that caused the touchscreen to break. I can't manually enter the passcode into the device to unlock it and be able to back up its contents onto my computer. Is there any way to unlock an iPhone through iTunes without using the actual phone?

This is impossible with a dark screen. Requires DFU mode that wipes it clean. But I found one trick that might work. No guarantees.
See the thread for a similar iPad issue. Best luck.

'iPad is disabled' fix without resetting using iTunes

Today I met my match with an iPad that had a passcode entered too many times, resulting in it displaying the message 'iPad is disabled - Connect to iTunes'. This was a student iPad and since they use Notability for most of their work there was a chance that her files were not all backed up to the cloud. I really wanted to just re-activate the iPad instead of totally resetting it back to our default image.
I reached out to my PLN on Twitter and had some help from a few people through retweets and a couple of clarification tweets. I love that so many are willing to help out so quickly. Through this I also learned that I look like Lt. Riker from Star Trek (thanks @FillineMachine).
Through some trial and error (and a little sheer luck), I was able to reactivate the iPad without loosing any data. Note, this will only work on the computer it last synced with. Here's how:
1. Configurator is useless in reactivating a locked iPad. You will only be able to completely reformat the iPad using Configurator. If that's ok with you, go for it - otherwise don't waste your time trying to figure it out.
2. Open iTunes with the iPad disconnected.
3. Connect the iPad to the computer and wait for it to show up in the devices section in iTunes.
4. Click on the iPad name when it appears and you will be given the option to restore a backup or setup as a new iPad (since it is locked).
5. Click 'Setup as new iPad' and then click restore.
6. The iPad will start backing up before it does the full restore and sync. CANCEL THE BACKUP IMMEDIATELY. You do this by clicking the small x in the status window in iTunes.
7. When the backup cancels, it immediately starts syncing - cancel this as well using the same small x in the iTunes status window.
8. The first stage in the restore process unlocks the iPad, you are basically just cancelling out the restore process as soon as it reactivates the iPad.

If done correctly, you will experience no data loss and the result will be a reactivated iPad. I have now tried this with about 5 iPads that were locked identically by students and each time it worked like a charm.