They are among the most iconic images of the Second World War—blurred, grainy and, the best of them, as stirring and in-the-moment as any battlefield photographs ever taken.

There are only 11 pictures—and nine surviving negatives—from that early morning of Tuesday, June 6, 1944, on Omaha Beach, the bloodiest of the D-Day landings, the one depicted in the movie Saving Private Ryan. But two of Hungarian photojournalist Robert Capa’s images, taken for the weekly Life magazine, stand out...