If your thermostat is like the one you pictured then it only controls heat. I would figure out how to open the thermostat and then unscrew it off the wall. Disconnect one of the two wires and touch it to the other wire. If the furnace comes on throw the cheap thermostat in the trash. Then drive over to Walmart and pick up a $25 digital thermostat and we'll help you wire it in. It will work better than the current one. It will be done way faster than warranty for another cheap mediocre factory stat. And it will be something cool that you installed yourself.
I installed mine a few years ago and I swear, out of the 30 minutes it took to install it, 20 - 25 minutes were spent making sure the mounting holes were level.
The link is the one I used but most of them will work. You just want to make sure it has a battery backup as the batteries are what is going to power the electronics.
Link