Here's the pin-out for a 24-pin ATX power connector: http://en.wikipedia.org/wiki/Computer_power_supply
If you'll notice the four additional pins are not merely 12v & ground, which are wht is put out by the 4-pin 12v power connector. Based on that alone, what you're proposing will not work.
You need a 20-to-24 pin adapter. Google for them. They're pretty cheap.