I'm not an economist, but here's an answer based on my understanding.
Suppose the market produces 10 widgets with 10 hours of labor. Those widgets are worth $1 each. Now, an innovation comes along that allows twice as many units to be produced with the same amount of labor.
The company can now produce 10 units with only 5 hours of labor. It then reallocates the five hours, either by assigning the workers to other products, dismissing them to find other work, or whatever.
Clearly, the economy is no less productive at this point. When the now surplus labor moves to another use, the economy is strictly more productive, producing more than before for the same amount of labor.
As you point out, if production is twice as efficient, the price will drop, and quantity demanded will increase at the new, lower, price. So, most likely, the company will wind up keeping the same amount of labor, but producing 20 units instead of 10.
It's true that all things being equal, the value of the 20 units will not be twice the value of the 10 units from before, since the price will drop substantially. So it's true that twice the productivity will not measure twice the monetary value.
However: the lower price produces deflation. That product deflates in price by around 50 percent, but in the overall US (say) economy, it amounts to a very small amount of deflation. Seeing that deflation, the Fed realizes it should increase the money supply (print more money) to keep the overall price level the same (or to keep it at its target 2% inflation, or whatever).
What happens, then, is: the price of widgets is lower, both before and after inflation, but the price of everything else is slightly higher to compensate.
If you add up the economy using all the old quantities but the new prices, they have to stay exactly the same, because inflation is zero. But: adding in the additional 10 widgets means that GDP (after inflation) has increased by their value, which means GDP is higher, with inflation at zero, and the same amount of labor.
In terms of goods produced, the country is obviously more productive after the innovation, because it has 10 more widgets with the same amount of labor. The monetary value might not show that -- it could indeed go down if the price of widgets falls enough. However, if you choose to measure in dollars instead of widgets, you have to adjust for inflation to keep the dollars constant. If you do that, you can prove mathematically that the overall value of everything produced must be higher. That's because the more the price of widgets drops, the more deflation you have, and the two cancel out, leaving only the value of the extra widgets.