This topic is one which has been of much debate; rather controversial truly. However we are certain that some ancients did in fact sail into the Atlantic; how far? That's the controversial debate. I will make a specific example of antiquated Greece, as that is the civilization which I hold primary specialty in; we have evidence that the Greeks traded with other peoples as far north as Britannia. The Straits of Gibraltar are at times referenced as, "The Pillars of Hercules/Heracles."; there are various myths which tell of heroes venturing beyond the Pillars. Typically, the myths regard the places beyond the Pillars as the ends of the Earth. It isn't likely that the majority of Greeks would see beyond the Mediterranean and the East; as I said however, we have archaeological evidence suggesting that there was trade with what is now Britain, to an extent. This of course is not very far into the Atlantic, but do bear in mind that I speak of the Greeks here, not of the Romans or Egyptians; perhaps other members may provide other information regarding a differing civilization? I can provide further information as well if this isn't satisfactory; I'll be sure to edit in a more detailed account when I have more time.

Further info: regarding the Romans, their campaigns in Britannia are widely known; to achieve this they must've sailed into the Atlantic to a point: the English Channel is a subdivision of the Atlantic, which serves as a passage from Gaul to Britannia, but is of course not far into the Atlantic as stated before. The Phoenicians furthermore also had trade with Britannia; I have a map of their trade routes if you're interested. Perhaps you notice a common theme here? We know certainly they ventured beyond the Mediterranean, but how far is debatable. What is certain is that they at least did venture far enough to reach Britannia and Hibernia (Ireland). Also, there are accounts of passage around Africa as well, not merely through the Straits. Again, I will edit in more information is this doesn't satisfy your inquiry.

Pliny the Elder (a Roman natural philosopher), to substantiate his claim that the Atlantic and Indian oceans are connected, gave several examples of expeditions (Punic and Greek) which had successfully sailed around Africa in both directions. On the other hand, he shares a story (although not necessarily with great confidence) that a ship of Indian merchants got caught in a storm, accidentally sailed around Asia, and washed ashore in Europe from the north, which seems less than likely, so one might not want to take his stories here without at least a healthy dose of doubt. In any case, his point was the geography, not the history of exploration.