The core meaning of the verb けす is not "turn off" but is "put off (a fire)" or "extinguish." The gas stove is a recent invention (I believe from the 19th century) and before that, people used to say things like "たきび の ひ を けして ください" and "かまど の ひ を けしてください (=please put out the fire in the oven/furnace)," and you still here "たばこのひをけしてください” and “ろうそくのひをけしてください”
So the expression in question doesn't actually mean "please turn off the fire of the gas." It's more like "please put off/extinguish the fire on the gas (stove)." If you think this is still too verbose, you can trim it into "ひをけしてください" by dropping the “gas” part, not the “fire” part.