Ahhhh, no. I'll leave it at that.
To answer the OPs question it will probably work, but if we are talking about an old 350W PSU from the P4 era, I'd get a new one. First, PSUs age and will eventually fail. Cheap units when they fail could sent a big voltage spike or bad ripple down the wires and possibly fry anything its connected to. Second, P4 era PSUs are not very efficient at what they do, so you're spending extra money on electricity that you shouldn't have to. I'd probably look for a nice new 350-400W PSU just so you don't have to worry about a possibly dying PSU killing your new card.
True but tell that to the few people out there that are still enjoying there Amiga or Dos era vintage IBM PCs.