The quasi-optimality criterion chooses the regularization parameter in inverse problems without requiring knowledge about the noise level. It is well known that this cannot yield convergence for ill-posed problems in the worst case. In this paper, we establish conditions providing lower bounds on the approximation error and the propagated noise error, such that these terms can be estimated from above and below by a geometric series. Using these we can show convergence and optimal-order error bounds for Tikhonov regularization with the quasi-optimality criterion both in the case of deterministic problems as well as for stochastic noise.