The total variation distance is a metric of central importance in statistics and probability theory. However, somewhat surprisingly, questions about computing it algorithmically appear not to have been systematically studied until very recently. In this paper, we contribute to this line of work by studying this question in the important special case of multivariate Gaussians. More formally, we consider the problem of approximating the total variation distance between two multivariate Gaussians to within an $ε$-relative error. Previous works achieved a fixed constant relative error approximation via closed-form formulas. In this work, we give algorithms that given any two $n$-dimensional Gaussians $D_1,D_2$, and any error bound $ε> 0$, approximate the total variation distance $D := d_{TV}(D_1,D_2)$ to $ε$-relative accuracy in $\text{poly}(n,\frac{1}ε,\log \frac{1}{D})$ operations. The main technical tool in our work is a reduction that helps us extend the recent progress on computing the TV-distance between discrete random variables to our continuous setting.