It's only not in your favor if you changed your card to cancel Netflix specifically, in which case you should've just logged in and canceled.
The simplest, safe route is to not give companies the newly updated number. If my Netflix lapses because I forgot to update the number after a card change (whatever the reason), they can email me, and then I will log in to my account and update the card on file.
You dispute the charge, just like any other unauthorized transaction. That's quite different than changing your card number under their feet, and will be received as such by Netflix.