We study the $d$-dimensional Vector Bin Packing ($d$VBP) problem, a generalization of Bin Packing with central applications in resource allocation and scheduling. In $d$VBP, we are given a set of items, each of which is characterized by a $d$-dimensional volume vector; the objective is to partition the items into a minimum number of subsets (bins), such that the total volume of items in each subset is at most $1$ in each dimension. Our main result is an asymptotic approximation algorithm for $d$VBP that yields a ratio of $(1+\ln d-χ(d) +\varepsilon)$ for all $d \in \mathbb{N}$ and any $\varepsilon > 0$; here, $χ(d)$ is some strictly positive function. This improves upon the best known asymptotic ratio of $ \left(1+ \ln d +\varepsilon\right)$ due to Bansal, Caprara and Sviridenko (SICOMP 2010) for any $d >3$. By slightly modifying our algorithm to include an initial matching phase and applying a tighter analysis we obtain an asymptotic approximation ratio of $\left(\frac{4}{3}+\varepsilon\right)$ for the special case of $d=2$, thus substantially improving the previous best ratio of $\left(\frac{3}{2}+\varepsilon\right)$ due to Bansal, Elias and Khan (SODA 2016). Our algorithm iteratively solves a configuration LP relaxation for the residual instance (from previous iterations) and samples a small number of configurations based on the solution for the configuration LP. While iterative rounding was already used by Karmarkar and Karp (FOCS 1982) to establish their celebrated result for classic (one-dimensional) Bin Packing, iterative randomized rounding is used here for the first time in the context of (Vector) Bin Packing. Our results show that iterative randomized rounding is a powerful tool for approximating $d$VBP, leading to simple algorithms with improved approximation guarantees.