We propose a novel formulation for stable sparse recovery from measurements contaminated by outliers and severe noise. The proposed formulation evaluates noise and outliers with a quadratic function and the minimax concave function, respectively, to reflect their statistical properties (Gaussianity and sparsity). This makes a significant difference from the conventional robust methods, which typically evaluate noise and outliers with a single loss function, leading to stability of the estimate. While the proposed formulation involves a nonconvex penalty to reduce estimation biases of sparse estimates, overall convexity of the whole cost is guaranteed under a certain condition by adding the Tikhonov regularization term. The problem is solved via a reformulation by the forward-backward primal-dual splitting algorithm, for which convergence conditions are derived. The remarkable outlier-robustness of the proposed method is demonstrated by simulations under highly noisy environments.