Barbarossa'a right, if you're trying to match "globalization" with what it is right now, vs. a comparable peak at 1914 for ocean trade and the British empire covering 25% of the globe or just long-distance trade networks. It's too easy to assume if well-documented recent Europeans weren't doing it, it didn't exist or matter.
Roman Empire has continual trade up into the Baltic Sea, England, North Africa, East Africa, Western India, and along the Silk Road to the Chinese coast (hence a Chinese embassy in ancient Rome), which at 3 continents and all of the major oceans seems pretty Global to me. We'll leave out remnants of Roman trade in the Americas like a whole ship at Rio De Jeneiro, amphorae, coins, armor, swords, etc..
The Phoenicians and Minoans are also reaching much vaster distances from Baltic amber and English copper and tin to Africa (i.e. Carthage but around much of it's coasts) to Western India. They also show up with a lot of Michigan copper and leave all sorts of physical traces in the Americas as well...and somebody had to be bringing the Egyptians Bolivian cocaine and N. American nicotine/tobacco for it to show up in hundreds of mummies over hundreds of years in ancient Egypt.
If globalization doesn't have to involve an ocean, which given the size of Eurasia and Africa it seems like it shouldn't, you've got domestic livestock, crops and fruit trees, all sorts of fabricated goods from flint arrowheads to Damascus blades, ideas, etc. traveling all over it in more frequent contact than we like to admit going back many thousands of years. Given land travel over vast distances was a lot harder than ocean travel, it should get more credit as the early globalization...we just don't have Tom Friedman's accounts about it to make it obvious (although Jared Diamond does a good job in "Guns, Germs, & Steel" and Jack Wetherford in his books on the Mongols' giant free trade zone or the Native Americans' continents-spanning trade networks.
Big, ocean-going ships that can move light, small high value cargos like manufactured goods or even bulk commodities like copper, timber, olive oil, grain, etc. go back many thousands of years so by some of these definitions you're referring more to global communications that are fast, so transatlantic telegraph lines and long-range radio if not world fiber optic networks and satellite communications and tracking.