I always ask this. If the spell wasn't cast, or if the save was made, would the damage happen? If so, it is spell damage. If not, it is it's own damage.
Just like sneak attack. Sneak attack damage isn't it's own damage source, it is an add-on amount on top of the normal attack damage, and even shares the same damage type (B/P/S) of the weapon used to perform the attack.
I would say that it works, with a good portion of that being that the spell level determines the damage of the burning each round, and also because to douse the flames uses the original spells DC. It is very clearly based on the spell as the source of the damage to me.
For clarity, Burning Magic does nothing for spells like Scorching Ray or Fiery Shuriken, as it specifies you have to fail a save in order to catch fire. So, only spells that allow a save, (and would negate the extra burn and the daze effect on a successful save) are ever going to be important.
The result of abilities that stack together to be very effective, just like many spells with saving throws. They are commonly more effective than attack roll spells, especially with both a class feature and a feat used to enhance them.