Even though casualties were astronomical, the D-Day invasion allowed the Allies to move into France. This was called the “Normandy Breakout,” but was more officially known as Operation Cobra. Eventually, France was liberated after the Battle of the Bulge. Retreating German forces crossed the border into Germany with Allied forces close behind. It was all defense for Germany from there. The Soviets beat Great Britain and the US to Berlin, finally defeating the Nazis.
It is not too strong of a statement to say that the victory on D-Day saved the world. Germany could no longer euthanize Jews, Slavs, homosexuals, the disabled and others they deemed undesirable. Global domination was becoming increasingly less possible. Attention could now be focused on attaining victory in the Pacific.