Kinetic models are widely used to simulate the dynamics of plasmas and rarefied gases across many application areas, including astrophysics, magnetically confined fusion, high-altitude aircraft, various spacecraft propulsion mechanisms, and microfluidic devices. Such models arise from a statistical-mechanics approximation of the underlying particle motion and typically take the form of nonlinear integro-differential equations defined on a high-dimensional phase space. In this work, we describe an approach known as micro-macro decomposition, which allows us to efficiently divide the kinetic equations into their low-order statistics and the remaining high-order corrections. A high-order discontinuous Galerkin finite element method is developed to numerically solve the resulting micro-macro system. A special treatment of the collision operator based on Gauss-Radau quadrature is used to achieve an asymptotic-preserving scheme in the high-collision limit. We consider different collision operators, including the Bhatnagar-Gross-Krook (BGK) and the Lenard-Bernstein operators. The resulting scheme is verified on several standard test cases.