This alternate version of Divergence theorem can be proved using Gauss-Green formula: $ \int_{\Omega} \partial_{x_i} u\, dx = \int_{\partial \Omega} u n_i dS \,, $ where $\boldsymbol{n} = (n_1,\ldots,n_k)$ is the unit outward normal vector to a smooth domain $\Omega\subset \mathbb{R}^k$.
Then let's consider this special case in $\mathbb{R}^3$. Let $\mathbf{A} = (A_1,A_2,A_3)$. The first component of $\nabla\times\mathbf{A}$ is $\partial_{x_2} A_3 - \partial_{x_3} A_2$, then: $ \int_{V} (\partial_{x_2} A_3 - \partial_{x_3} A_2)dV = \int_{\partial V} (n_2 A_3 - n_3 A_2)dS, $ in which the right hand side is exactly the first component for $\mathbf{n}\times \mathbf{A}$. The second and the third components are proved in the same way.