Consider the elliptic operator given by $$ \mathscr{L}_εf= {b} \cdot \nabla f + εΔf $$ for some smooth vector field $ b\colon \mathbb R^d \to\mathbb R^d$ and a small parameter $ε>0$. Consider the initial-valued problem $$ \left\{ \begin{aligned} &\partial_ t u_ε= \mathscr L_εu_ε,\\ &u_ε(0, \cdot) = u_0(\cdot), \end{aligned} \right. $$ for some bounded continuous function $u_0$. Denote by $\mathcal M_0$ the set of critical points of $b$ which are stable stationary points for the ODE $\dot x (t) = b (x(t))$. Under the hypothesis that $\mathcal M_0$ is finite and $ b = -(\nabla U + \ell)$, where $ \ell$ is a divergence-free field orthogonal to $\nabla U$, the main result of this article states that there exist a time-scale $θ^{(1)}_ε$, $θ^{(1)}_ε\to \infty$ as $ε\rightarrow 0$, and a Markov semigroup $\{p_t : t\ge 0\}$ defined on $\mathcal M_0$ such that $$ \lim_{ε\to 0} u_ε(tθ^{(1)}_ε, x) =\sum_{m'\in \mathcal M_0} p_t(m, m')\, u_0( m'), $$ for all $t>0$ and $ x$ in the domain of attraction of $m$ for the ODE $\dot{x}(t)= b( x(t))$. The time scale $θ^{(1)}$ is critical in the sense that, for all time scale $\varrho_ε$ such that $\varrho_ε\to \infty$, $\varrho_ε/θ^{(1)}_ε\to 0$, $$ \lim_{ε\to 0} u_ε(\varrho_ε, x)=u_0(m) $$ for all $x \in \mathcal D(m)$. Namely, $θ_ε^{(1)}$ is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [Landim, Lee, Seo, forthcoming] we extend this result finding all critical time-scales at which the solution $u_ε$ evolves smoothly in time and we show that the solution $u_ε$ is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of $b$.