We study a class of Lotka-Volterra stochastic differential equations with continuous and pure-jump noise components, and derive conditions that guarantee the strong stochastic persistence (SSP) of the populations engaged in the ecological dynamics. More specifically, we prove that, under certain technical assumptions on the jump sizes and rates, there is convergence of the laws of the stochastic process to a unique stationary distribution supported far away from extinction. We show how the techniques and conditions used in proving SSP for general Kolmogorov systems driven solely by Brownian motion must be adapted and tailored in order to account for the jumps of the driving noise. We provide examples of applications to the case where the underlying food-web is: (a) a 1-predator, 2-prey food-web, and (b) a multi-layer food-chain.