The sum of Log-normal variates is encountered in many challenging applications such as in performance analysis of wireless communication systems and in financial engineering. Several approximation methods have been developed in the literature, the accuracy of which is not ensured in the tail regions. These regions are of primordial interest wherein small probability values have to be evaluated with high precision. Variance reduction techniques are known to yield accurate, yet efficient, estimates of small probability values. Most of the existing approaches, however, have considered the problem of estimating the right-tail of the sum of Log-normal random variables (RVS). In the present work, we consider instead the estimation of the left-tail of the sum of correlated Log-normal variates with Gaussian copula under a mild assumption on the covariance matrix. We propose an estimator combining an existing mean-shifting importance sampling approach with a control variate technique. The main result is that the proposed estimator has an asymptotically vanishing relative error which represents a major finding in the context of the left-tail simulation of the sum of Log-normal RVs. Finally, we assess by various simulation results the performances of the proposed estimator compared to existing estimators.