Alright, so this question is a little bit old, but I'd been looking for a similar result for a long time and I found it the other day. Daniel Stroock has a complete proof of the divergence theorem in two of his books, "A Concise Introduction to the Theory of Integration" and "Essentials of Integration Theory for Analysis." As it turns out, the result in the first book is somewhat more general (fewer assumptions on f), but the proof is simpler in the second. The result states:
Again let $G$ be a smooth region in $\mathbb{R}^N$ and $U$ and open neighborhood of $\bar{G}$. If $F:U\rightarrow\mathbb{R}^N$ is continuously differentiable and either $G$ is bounded or $F\equiv 0$ off a compact subset of $U$, then
$\int\limits_G div F(x) dx = \int\limits_{\partial G} (F(x),n(x))_{\mathbb{R}^N} \lambda_{\partial G}(dx)$
In order to get integration by parts, let $F=fg$ for $f:\mathbb{R}^N\rightarrow \mathbb{R}^N$ and $g:\mathbb{R}^N\rightarrow\mathbb{R}$ and then apply the chain rule.