Follow up question, how far back were Old World ships capable of crossing the Atlantic? What about East-asian ships and the Pacific?

Columbus' voyage has little to do with naval technology. European ships were capable of reaching the New World by the year 1000 or so (or earlier), which is the date of the settlement at L'Anse aux Meadows in Newfoundland.

Adapted from an earlier answer:

Columbus' discovery of the Americas has very little to do with technology, actually. The Norse settled in Greenland for a couple of centuries and apparently also spent some time in L'Anse aux Meadows, in Newfoundland. They made that crossing in the tenth century, most likely around the year 980, and there's no particular technological reason why they couldn't have kept on exploring down the American coast, then returning up the coast via Greenland and Iceland, and then on to the northern UK and Europe. Their ships used square sails and wind power in part, but were also light enough to row for long distances when the wind failed or wasn't blowing in the right direction.

There are a few theories on why the Greenland Norse settlement was abandoned, and I'm not particularly qualified to comment on those, but you may find some answers in this section of our FAQs.

Anyhow, to get to Columbus: The main reason he thought the voyage across the Atlantic would be practicable is that he thought the earth was much smaller in circumference than it is -- I've written about this more here -- and that he had some knowledge of the circular wind patters of the Atlantic, though it's not clear where he got those from.

In Columbus' time, Portuguese explorers had been using carracks and caravels to explore the African coast; those ship types were handier in sailing upwind than earlier types. But early exploration of Africa was limited by the stowage of ships, and how long they could remain at sea without re-provisioning; because sailing upwind took much longer than downwind, the return voyage was a constraint.

In Columbus' first voyage in 1492, he made it across the Atlantic to the Bahamas in about five weeks, following the prevailing westerly winds; he was able to return home by turning north and using the prevailing northeasterly winds and currents off the American coast, which turn east, to return to Europe by a more northerly route. The pattern of currents was known to Portuguese explorers, though the precise source of this is unclear.

Here's another similar question about Columbus and technology; that answer is here.

Basically, I would advise you to be extremely skeptical of technology-driven arguments for why people do a thing: most often, we decide we need to do a thing and invent our way to it. (That's not to say that technology doesn't have knock-on effects; it is to say that we didn't just build ships capable of crossing the Atlantic and then say "yay now we can explore Ocean tiles.")

A follow up question. Are you excluding Vikings in your question?Given that there is solid evidence that the first Europeans to sail to North America were Vikings who set up their first camp (excluding Greenland) in 1001, that sets a date much earlier than 1392.