We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $\pi$ on $\mathbb{R}^d$, based on (overdamped) Langevin diffusions. This method inspired by [PP18] and [GMS20] relies on a multilevel occupation measure, i.e. on an appropriate combination of R occupation measures of (constant-step) Euler schemes with respective steps $\gamma_r=\gamma_0 2^{-r}$, r=0,...,R. We first state a quantitative result under general assumptions which guarantees an epsilon-approximation (in a $L^2$-sense) with a cost of the order $\epsilon^{-2}$ or $\epsilon^{-2}|\log \epsilon|^3$ under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential $U:\mathbb{R}^d\rightarrow \mathbb{R}$ and obtain an $\epsilon$-complexity of the order $O(d \epsilon^{-2}\log^3(d \epsilon^{-2}))$ or $O(d \epsilon^{-2})$ under additional assumptions on $U$. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by ${\bar{\lambda}_U\vee 1)^2}{\underline{\lambda}_U^{-3}} d\varepsilon^{-2}$ (where $\bar{\lambda}_U$ and $\underline{\lambda}_U$ respectively denote the supremum and the infimum of the largest and lowest eigenvalue of $D^2U$).\smallskip We finally complete these theoretical results with some numerical illustrations including comparisons to other algorithms in Bayesian learning and opening to non strongly convex setting.