In this paper we investigate connections between minimal Lagrangian submanifolds and holomorphic vector fields in K\"ahler manifolds. Our main result is: Let $M^{2n}$ $(n \geq 2)$ be a K\"ahler-Einstein manifold with positive scalar curvature with an effective, structure-preserving action by an $n$-torus $T^n$. Then precisely one regular orbit $L$ of the $T^n$-action is a minimal Lagrangian submanifold of $M$. Moreover there is an $(n-1)$-torus $T^{n-1} \subset T^n$ and a sequence of non-flat immersed minimal Lagrangian tori $L_k$ in $M$ such that all $L_k$ are invariant under $T^{n-1}$ and $L_k$ locally converge to $L$ (in particular the supremum of the sectional curvatures of $L_k$ and the distance between $L$ and $L_k$ go to $0$ as $k \mapsto \infty$). This result is new even for $M=\mathbb{C}P^n$ for $n \geq 3$.