At my international relation class today, my professor stated that it was an European war until the US got involved. This seemed strange to me, since I believe it's fairly common knowledge that the Great Powers of the time were colonialist, thus their colonies got involved since the beginning. Is his statement correct?
No.
There are two definitions of a "world war," I suppose. It could be geopolitical - does it involve powers from all over the world? Or it could be geographic - does it take place all over the world? By either definition, the Great War was a World War by 1914.
To begin with, the conflict wasn't only European, nor did it involve only European Powers. Japan, for example, was actively involved in the war from 1914 to 1918, taking German possessions in Asia like the treaty port of Tsingtao in China. There was also the fighting in the Near East take took place between the Allies and the Ottoman Empire, which was a mostly Asian empire. Furthermore, when it comes to North American participation, the Canadians might object to being considered "European." Not to mention the participation of troops from Bermuda, Australia, New Zealand, South Africa, and other regions that were closely-aligned with the British, but had a certain degree of independence or autonomy.
Furthermore, the war was global in scope. There were naval battles in the Falkland Islands near Argentina. There were extensive campaigns in sub-Saharan Africa. There was campaigning across vast sections of Eastern Europe. There was bitter fighting in the Middle East and what was then Palestine.
In fact, the global scale of the war is what draws the U.S. into the conflict. Unrestricted submarine warfare in the Atlantic (essentially sinking merchant ships without warning) and the Zimmerman telegram (a German effort to bring Mexico into the war on the German side) were major factors in why the Americans entered the war.