Quote: This statement is not necessarily true.
It takes a certain measure of power to push a given object along at a given speed against normal rolling resistance & against the air.
For the purpose of simplicity, lets just say that in 4th gear you are doing 4,000rpm
At this speed you have the throttle kicked open about 1/3rd & the engine develops X amount of power.
Now we slip it into OD 5th & the revs drop to 3,000 rpm [for example]. If the cylinders are still operating at the same measure of volume fill for each intake stroke, then at that reduced engine speed, we would have less power & the car would slow down since we are now burning less fuel [per minute] than before.

Since the cylinders now have more time to receive a fresh charge of fuel, assuming the same throttle setting as before, then more fuel [& air] will flow in, giving each intake stroke a greater volume, but for each time unit, like a whole minute, we have fewer intake strokes. [lower rpM]
So now we have fewer intake strokes with more fuel/air per stroke. This in itself may not produce the required power at this reduced engine speed to maintain road speed & if this proves to be true, then we must open the throttle a little more to bring the power back up to the previous figure of X, but at the new lower engine speed in order to maintain our previous cruise speed.
So now we have even more fuel per intake stroke at our reduced rpm levels.

Simply put, we must burn fuel to make power. Each engine design has its own power-to-engine-speed curve for any given throttle opening & we must remember that as our vehicle speed increases, so does the air resistance, at the rate of the square of speed increase. [eg. multiply the speed by 2 & square that number [2x2=4] for increase of air resistance]

In the end, we may not be getting a fuel economy improvement at all, but instead, at certain speeds, we may be causing our engines to run below their designed point of greatest efficiency & doing both it, & ourselves a disservice.

Even the claim of reduced wear may not always be true since the load at the reduced engine speed is greater, so once again, there are fewer crank rotations, fewer piston reciprocations, but each one is done at a higher loading.

If you can engage 5th gear & still have the engine 'in the zone' of maximum efficiency, then it's all good, but pull 5th gear when too slow, or when battling a headwind [or both] & the economy thing can easily go straight out the window.


So what was the point of all this?
Do not assume that the presence of an overdrive gear will automatically provide a reduction in fuel consumption or wear rates in all instances at all times as even the engaging of the overdrive gear involves the use of indirect gearing which immediately drives up the drag through the gearbox itself which imposes a small additional power burden on the engine.
Knowledge is power. Get some.