Connecting a 3.5mm audio jack to a 12V DC power cable
What you want to do should work, just as you mentioned, you need to make sure the polarity is correct.
Looking at the device, your center conductor should be +. 
So, just get your 12V power supply, cut the connector off, strip the wires and check the voltage with a multimeter. If you read positive +12V, you have the same polarity as the multimeter (common to common).