On a previous car I bought which finance was owed I done the following.
Got seller to get a settlement figure in writing.
In my case it was through PTSB.
I went to my local PTSB with the seller and paid the finance off, holding onto the receipt.
Gave the seller the balance.
Drove home in shiney new car

I think I may have had to call the finance house just before going the bank to get their AC and NSC details and simply put the loan ref. number in the subject field when doing the transfer.