Short Answer:

Your question presumes that rowed ships disappeared-- they didn't. They were not particularly well suited to long distance blue water navigation, so you see the cog and ships derived from that design (carracks, caravel, nau, etc) used for those purposes.

Discussion:

Rowed galleys were common in the Mediterranean through the Renaissance; the battle of Lepanto in 1571 was fought by fleets of rowed galleys, some 200 and more on each side. These galleys could put up a lateen or square sail, but in battle they were rowed, and a veteran of Salamis (480 BCE, roughly 2000 years earlier) would recognized the ships as quite similar to triremes, though the gunpowder weapons would have been new.

Rowed vessels are challenging to handle in open ocean and require a lot of manpower; they're also necessarily shallow drafted with a low freeboard (the distance from the waterline to the deck), precisely the opposite of what you'd want to be in sailing around the world in rough seas and carrying cargo. The cog/carrack does far better, carrying more cargo, more robustly and in the hands of skilled sailors can take on challenging weather and seas. Developed in Northern Europe, this technology spreads south to Portugal and the Mediterraneans in the later middle ages. While these were better suited to blue water routes, travel to the New World, India, the Phillipines-- galleys still were found into the 18th century in the Mediterranean.

France in particular used galleys as a judicial punishment, a practice that goes back to antiquity

The ledgers of the galleys of France held in the French naval archives at Toulon contain particulars of 60,401 men who had been sentenced between 1660 and 1748 to serve as captives chained to the rowing benches.

-- by the late date of 1748 we read that there are still some 5,000 men condemned "to the galleys", though not many would be pulling an oar by that time. Most would have been ordinary prisoners, the prisoners having become a more general supply of forced labor. The "galley" came to be as much a symbol of the repression of the Old Regime, a bit like the Bastille in unhappy memory. The Republic of Venice maintained some galleys into the late 18th century, and there was still occasional use of oared fighting ships in the 19th.

It should be noted that the combination that is seen in the Mediterranean - fighting galleys and sailed merchant ships- was also seen in classical times. The Romans had large sailing vessels, capable of long distance journeys, for example trade between the Red Sea ports and Sri Lanka. What the Romans didn't have was an ability to navigate confidently out of sight of land, so that limited their exploration; lack of compass, sextant and maps was more limiting than the naval architecture; while most ships were likely of the 100 ton class, underwater archaeology now gives us confidence that sail powered ships of 500 tons and perhaps more existed.

Sources

Ó hANNRACHÁIN, EOGHAN. “MEN OF THE WEST IN THE GALLEYS OF FRANCE.” Journal of the Galway Archaeological and Historical Society, vol. 59, 2007, pp. 37–45. JSTOR, www.jstor.org/stable/20696599.

Royal, Jeffrey G., and John M. McManamon. “Three Renaissance Wrecks from Turkey and Their Implications for Maritime History in the Eastern Mediterranean.” Journal of Maritime Archaeology, vol. 4, no. 2, 2009, pp. 103–129. JSTOR, www.jstor.org/stable/23747343.

Guilmartin J Jr (1974) Gunpowder and galleys: changing technology and Mediterranean warfare at sea in the sixteenth century. Cambridge studies in early modern history. Cambridge University Press, Cambridge

Bamford, Paul W. Fighting Ships and Prisons: the Mediterranean Galleys of France in the Age of Louis XIV. (Minneapolis:1973)