1744, "public room for music or dancing," from Italian casino, literally "a little house," diminutive of casa "house," from Latin casa "hut, cottage, cabin," which is of uncertain origin. The card game (also cassino) is attested by that name from 1792. Specifically as "building for aristocratic gambling" by 1820, first in an Italian context.
[T]he term Casino [is] indiscriminately applied to a set of farm offices, a country-seat, a gambling house, and a game of cards ... [Jane Waldie Watts, "Sketches Descriptive of Italy in the Years 1816 and 1817," London 1820]