Yes it saves fuel, unless you get really slow (well below cruise speed - more later.)
Since drag (and thus power required) increases with the square of velocity it will save fuel for the total flight. A simple example even if not truly realistic: going twice as fast generates four times the drag and thus four times the fuel burn (ignoring small changes in engine efficiency) and it gets you there in half the time, so it would still cost twice the fuel burn for the flight. And as a side note, drag is proportional to air density. So flying higher reduces drag and fuel burn for the same speed.
You are correct. There is an optimal air speed. It's called Max Endurance speed. It is at the point where total drag (induced and parasitic) is at a minimum. It is significantly slower than normal cruise speeds. Going slower than that will increase power required and fuel burn even as you go slower. This is 'operating on the backside of the power curve.'
But for maximum range, that occurs slightly higher than minimum power speed at the speed for best L/D. Going slower than the max range speed will result in more fuel over a fixed distance.
One caveat is that if you have a head wind or tail wind, the fuel burn for a given distance will change.
And to make things more interesting; the airlines look at total costs for the flight, not just fuel burn. Crew costs are also significant. In the performance computations within the FMS there is an input called 'Cost Index.' Each airline calculates the index (offline) and provides it to the crew for input. It basically provides an adjustment factor based on the relative costs of fuel vs. crew. With high relative fuel costs, the Cost Index will calculate a slower speed to save fuel. If fuel is cheap relative to the crew, the CI will calculate a higher speed to reduce the crew costs (since they get paid by the hour.)