I think your skepticism may come from not understanding the process behind the claim.
A single unit of blood is separated into 4 main "blood products": red blood cells, plasma, platelets, and white blood cells. Another product called cryoprecipitate can be produced from frozen then thawed plasma and is used in special circumstances. Whole blood is rarely used for transfusions anymore because of problems with transfusion reactions and, quite frankly, except in the case of massive hemorrhage, a single person rarely needs all these components all at once (and even then they probably won't need white blood cells). Red blood cells (or packed red blood cells) are what most people think of when they get a "blood transfusion." Plasma is given to people who do not have enough clotting factors in their blood to stop bleeding that is currently occurring or if it should occur. Platelets are given to people who are not producing enough platelets to keep them from bleeding to death. White blood cells are rarely given anymore, but there may be occasion to use them in specific cases.
Now did that one unit save up to 3 lives? Well, to answer that the question of whether their lives were in danger to begin with has to be answered. The answer is, "Yes!" When will they die without the product? For some it's in the next several minutes, for others it may be hours, days or weeks, but in every case these peoples lives are in danger without the products derived from human blood.
But did that unit all by itself achieve this? No, but in concert with the donations of others it contributed to saving "up to 3 lives." Because doctors try avoid using precious blood products until it is absolutely necessary by treating patients with other methods if possible, so by the time a person definitely requires blood products, they will get more than one unit, but had they been given products earlier without giving the 'other methods' a chance to work they would have needed just as much, maybe more, over time.
The American Cancer Society has an excellent summary of types of blood transfusions and what they are used for here.
I don't think the claim is derived from any particular set of statistics or the hard and fast numbers that you are seeking, but from the process involved.
EDIT:
The best collection of statistical information online may be Blood Transfusion and Donation published by the National Institutes of Health.