My state has a reciprocity agreement with IL, but I paid state taxes in both states. How do I receive a credit for the state taxes I paid to IL?

You don't take a credit, you file an Illinois Non-resident state return to get the tax that was withheld refunded. 

I have set up my IL state return on TurboTax but it runs through the whole process without mentioning that. It says I still owe an additional small amount of IL state tax.

There is still income allocated to IL.  You might start with viewing the IL Tax Summary to help you narrow down which income source is still showing for IL.  Below are the steps to see the IL summary.

 

Next, be sure in the IL interview that you are in the nonresident interview to allow the allocation process.  

 

Then, you will need to carefully go back through the IL nonresident return interview specifically the "Income/Adjustments" section. You must allocate zero (0) income to IL from all sources so that there will be no IL taxable income. 

 

1. Select "Tax Tools" in the left hand menu
2. Select "Tools"
3. Select "View Tax Summary"
4. Select "Preview My 1040" in the left hand menu

5. Select "IL Tax Summary"