In this paper we focus on the solution of online problems with time-varying, linear equality and inequality constraints. Our approach is to design a novel online algorithm by leveraging the tools of control theory. In particular, for the case of equality constraints only, using robust control we design an online algorithm with asymptotic convergence to the optimal trajectory, differently from the alternatives that achieve non-zero tracking error. When also inequality constraints are present, we show how to modify the proposed algorithm to account for the wind-up induced by the nonnegativity constraints on the dual variables. We report numerical results that corroborate the theoretical analysis, and show how the proposed approach outperforms state-of-the-art algorithms both with equality and inequality constraints.