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We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form $-a:\nabla \nabla u+b·\nabla u+cu=f\left(x\right)$, $x\in \Omega ={(0,1)}^d\subset {ℝ}^d$, where $a\in {ℝ}^{d\times d}$ is a symmetric positive semidefinite matrix, using piecewise polynomials of degree ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution and its stabilized sparse...