We investigate the existence of sign-changing solutions for Kirchhoff-type problems with $p$-Laplacian involving critical exponent: \begin{equation*} -\left(1+b \vert \nabla v \vert_p^p\right)\Delta_p v+a(x)\vert v\vert^{p-2}v=\vert v\vert^{p^*-2}v+\lambda f(v), \quad x\in \mathbb{R}^N, \end{equation*} where $b$ and $\lambda$ are positive parameters, $\Delta_p v= \operatorname{div}(\vert \nabla v\vert^{p-2} \nabla v)$, $p^*=\frac{Np}{N-p}$, $1