In this paper, we consider linear systems in input-output form and introduce a new adaptive linear quadratic Gaussian (LQG) control scheme which is shown to be self-optimizing. The identification algorithm incorporates a cost-biasing term, which favors the parameters with smaller LQG optimal cost and a second term that aims at moderating the time-variability of the estimate. The corresponding closed-loop scheme is proven to be stable and to achieve an asymptotic LQG cost equal to the one obtained under complete knowledge of the true system (self-optimization). The results of this paper extend in a nontrivial way previous results established along the cost-biased approach in other settings.