Learning the governing equations in dynamical systems from time-varying measurements is of great interest across different scientific fields. This task becomes prohibitive when such data is, moreover, highly corrupted, for example, due to the recording mechanism failing over unknown intervals of time. When the underlying system exhibits chaotic behavior, such as sensitivity to initial conditions, it is crucial to recover the governing equations\ with high precision. In this work, we consider continuous time dynamical systems $\dot{x} = f(x)$ where each component of $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^d$ is a multivariate polynomial of maximal degree $p$; we aim to identify $f$ exactly from possibly highly corrupted measurements $x(t_1), x(t_2), \dots, x(t_m)$. As our main theoretical result, we show that if the system is sufficiently ergodic that this data satisfies a strong central limit theorem (as is known to hold for chaotic Lorenz systems), then the governing equations $f$ can be\ exactly recovered\ as the solution to an $\ell_1$ minimization problem---even if a large percentage of the data is corrupted by outliers. Numerically, we apply the alternating minimization method to solve the corresponding constrained optimization problem. Through several examples of three-dimensional chaotic systems and higher-dimensional hyperchaotic systems, we illustrate the power, generality, and efficiency of the algorithm for recovering governing equations from noisy and highly corrupted measurement data.