In this letter, we optimize the channel estimator of the cascaded channel in an intelligent reflecting surface (IRS)-assisted millimeter wave (mmWave) multi-antenna system. In this system, the receiver is equipped with a hybrid architecture adopting quantized beamforming. Different from traditional multiple-input multiple-output (MIMO) systems, the design of channel estimation is challenging since the IRS is usually a passive array with limited signal processing capability. We derive the optimized channel estimator in a closed form by reformulating the problem of cascaded channel estimation in this system, leveraging the typical mean-squared error (MSE) criterion. Considering the presence of possible channel sparsity in mmWave channels, we generalize the proposed method by exploiting the channel sparsity for further performance enhancement and computational complexity reduction. Simulation results verify that the proposed estimator significantly outperforms the existing ones.