There's really no way to say for certain whether it would or wouldn't work. You'd have to simply plug it in and try it out to see if it goes or not. It works for some people and doesn't for others.
But even if it does seem to work initially, I would recommend purchasing an ATX 2.0 power supply some time in the future to replace the generic power supply that came with your case. If you're really trying to save a little coin right now, then that's cool - just buy the adapter and see what happens. If it works, you've got a solution until such time as you can afford to upgrade to an ATX 2.0 supply (if you want to). If it doesn't work, well...you know what to do.