Well, not sure what LCD you've got, but I've got a dell 2405fpw, and it works exactly like you'd expect.
If you power it up, then UNPLUG the power cable (or in my case, hook the power cable for it up to a smart strip), the smart strip will completely shut it off when I power down the pc.
And when I turn on the PC, the smartstrip restores power to the LCD, which immediately powers on, no button pushing required.
But other lcd's might not work that way.