This paper reports an extended state observer (ESO)-based robust dynamic surface control (DSC) method for triaxial MEMS gyroscope applications. An ESO with non-linear gain function is designed to estimate both velocity and disturbance vectors of the gyroscope dynamics via measured position signals. Using the sector-bounded property of the non-linear gain function, the design of an $\mathcal{L}_2$-robust ESO is phrased as a convex optimization problem in terms of linear matrix inequalities (LMIs). Next, by using the estimated velocity and disturbance, a certainty equivalence tracking controller is designed based on DSC. To achieve an improved robustness and to remove static steady-state tracking errors, new non-linear integral error surfaces are incorporated into the DSC. Based on the energy-to-peak ($\mathcal{L}_2$-$\mathcal{L}_\infty$) performance criterion, a finite number of LMIs are derived to obtain the DSC gains. In order to prevent amplification of the measurement noise in the DSC error dynamics, a multi-objective convex optimization problem, which guarantees a prescribed $\mathcal{L}_2$-$\mathcal{L}_\infty$ performance bound, is considered. Finally, the efficacy of the proposed control method is illustrated by detailed software simulations.