The long-wavelength properties of a noisy Kuramoto-Sivashinsky (KS) equation in 1 + 1 dimensions are investigated by use of the dynamic renormalization group (RG) and direct numerical simulations. It is shown that the noisy KS equation is in the same universality class as the Kardar-Parisi-Zhang (KPZ) equation in the sense that they have scale invariant solutions with the same scaling exponents in the long-wavelength limit. The RG analysis reveals that the RG flow for the parameters of the noisy KS equation rapidly approach the KPZ fixed point with increasing strength of the noise. This is supplemented by numerical simulations of the KS equation with a stochastic noise, in which scaling behavior close to the KPZ scaling can be observed even in a moderate system size and time.