I was messing around trying to make a single transistor and resistors do this with a self powered circuit.
R3 and R4 create a voltage divider, sampling a section of the supply voltage. When the supply is at 12V (normal), then R4 is dropping enough voltage to turn on Q1. This shorts out R1 to ground, preventing the current from flowing through the LED and R2.
When the supply drops to 11V, R4 no longer drops enough voltage to turn on Q1, so the current flows through R1, D1 and R2, lighting the LED.
In reality this may prove too inefficient, too picky about resistor values and transister gain. I just thought it would be fun to try and make it work. At the very least you'd need a trim pot near R3 and R4.
As someone else said, a Zener would be a better idea. I was just going for as simple as I could here.
Edit: Green is the input voltage, Blue is LED current. I just used a sine wave to make it easy to test.