In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equation $$u_{tt}-M({parallel}{ abla}u{parallel}^2){ riangle}u+{alpha}u_t+f(u)=0;in;{Omega}{ imes}[0,{infty}),\u(x,t)=0;on;{Gamma}_1{ imes}[0,{infty}),\{frac{{partial}u}{partial{ u}}}+g(u_t)=0;on;{Gamma}_0{ imes}[0,{infty}),\u(x,0)=u_0,u_t(x,0)=u_1;in;{Omega}$$ with nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some difficulties due to the presence of nonlinear terms $M({parallel}{ abla}u{parallel}^2)$ and $g(u_t)$ by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.