This is the first world map ever made. It was created by Al-Idrisi, a Moroccan Muslim geographer who was in the court of Roger II (king of Sicily) in 1154. As you can see, the first world map ever happens to be the other way around than later maps, which isn't necessarily wrong as there isn't any up or down in outer space.
And before anyone starts to say anything about maps made by ancient Greeks, let me tell you that no such thing exists. Ancient Greek geographers merely described different regions in their literature, they never drew maps. Any such maps were made by new age Europeans centuries after Al-Idrisi made his map.
(PS. I expect Australians to like maps like this, because then they don't have to be down under. :P)
By 1154 there were many maps.
It is said that the Greek Anaximander made the first map (as far as recorded history is concerned) but the map is now lost.
However, Ptolemy made a map of the world in the second century. It's known as Ptolemy's world map. The originals no longer exist though. There's also the Babylonian Map of the World (on a clay tablet). There is also the T and O map which dates back to 1050. Ibn Hawqal (another Arab scientist) also created a map a few decades before Idrisi.
Romans made maps also alongside the Chinese long before Al-Indrisi.
Al-Idrisi simply produced a map which was the most accurate for his time and was for a few centuries. He wasn't the first to make a world map though and earlier ones exist (even if not as detailed).
The more correct term would be, map of Al-Idrisi (first most detailed world map in history).