...the American Revolution was also a world war. With the American victory at Saratoga in 1778, France entered the war on the American side. The French wanted to avenge its defeat in 1763 at the hands of the British in the Seven Years' War. It had been secretly supplying the Americans with military supplies since 1775 awaiting an opportunity to side openly with the revolting Americans. By 1780, both Holland and Spain joined the French and Americans. (The Spanish, it is true, were a little hesitant to make war against another colonial power, but the possibility of destroying British trade hegemony was too powerful to resist. The Spanish monarchy would regret its decision in the nineteenth century when its own colonies would revolt citing the American example). With their seafaring fleets, America's European allies attacked British possessions in the West Indies, Africa, and India, thus spreading the war over the face of the globe.
See also "The First Salute" by Barbara Tuchman.
A better case can be made that the Seven Years War was the first world wide war.