We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving single-phase incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss-Lobatto-Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge-Kutta method. We present a consistent boundary treatment allowing us to use both central and Lax-Friedrichs fluxes for the numerical flux in the discontinuous Galerkin approach. We present two benchmark cases: lid-driven cavity flows for Re=400-5000 and flows around an impulsively started cylinder for Re=550-9500. Computational results are compared with those of other theoretical, experimental, and computational work that used a multigrid method, a vortex method, and a spectral element model.