This sounds like solid evidence that the Devil is scrupulously fair! He could have just said that his performance was better; his demon backup band would surely have taken his side. I suspect he enjoys the challenge of outdoing mortals, and thus allows them a sporting chance of escape. Note the impressive loophole in this contract! [0]
Do we ever see proof of Johnny's lack of faith in God, though? I suppose he's fairly prideful, by way of his assertion about being the best there ever was, but I don't think it's as clear as that. He does beat the Devil in a fiddle contest; is it prideful if it's a justified statement?
The Devil clearly can't just take souls without their owner's consent, so he needs to project an air of trustworthiness. There's a serious power asymmetry between mortals and the Devil, and the only higher authority one could think of is God himself. One suspects that God wouldn't easily take the Devil's side, so it behooves (heh) the Devil to be seen as honourable as possible, to ensure smooth and continued soul acquisition.
Johnny's lack of faith is that he doesn't invoke the name of God. The devil has God's absolute power when applied to testing humanity. The devil lost because they wanted to. By losing he has planted the seeds in Johnny that he can live without God's help.
It is possible the author intended something else, but this interpretation is the closest to Catholic doctrine I have seen. IIRC I am remembering it from a priest's comments on the song.
"Those who study demonology quickly realize that any power lies with the demons." - Some DND manual.
[0] https://news.ycombinator.com/item?id=29127547