Would this PCIe 2.1 Video Card Work on My PCIe 1 Motherboard?
Yes, PCI 2.1 is backwards compatible with PCIe 2.0/1.0. While it will run slightly slower, it will only be a few FPS.
http://en.wikipedia.org/wiki/PCI_Express
It shouldn't have a problem as PCIe 2.0/2.1 is backwards compatible with 1, but it won't operate at full capacity.
Edit: meaning, it won't be as fast as it would in a PCIe 2.1 or 2.0 slot. I doubt you'd notice much of a difference though.