The celebrated Brezis-Ekeland principle [C. R. Acad. Sci. Paris Ser. A-B, 282 ( 1976), pp. Ai, A1197-A1198, Aii, and A971-A974] characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems. This note extends this characterization to doubly nonlinear evolution equations driven by convex potentials. The characterization is exploited in order to establish approximation results for gradient flows, doubly nonlinear equations, and rate-independent evolutions.