The "dark side of the moon" is a bit of a misnomer. The moon is tidally locked to Earth, meaning the same side always faces us. The other side of the moon *is* illuminated by the sun, it just isn't visible from Earth.

So, the first people to see the "dark side" of the moon were the Soviet Union's Luna 3 probe in 1959. It took the first pictures of the far side of the moon and sent them back to Earth.