Surrogate-assisted multiobjective evolutionary algorithms (SAMOEAs) are a promising approach for solving expensive multiobjective optimization problems (EMOPs), wherein the number of function evaluations is extremely restricted due to expensive-to-evaluate objective functions. However, most SAEAs are not well-scaled to high-dimensional problems because the accuracy of surrogate models degrades as the problem dimension increases. This paper proposes a dimensionality reduction-based SAEA, which involves the following two strategies to address high-dimensional EMOPs. First, mapping high-dimensional training samples to a low dimensional space in building surrogate models can boost the accuracy of surrogate models. Second, compared to approximation-based surrogate models, reliable classification-based models can be obtained under a few training samples. Accordingly, the proposed algorithm is designed to integrate a dimensionality reduction technique into an existing classification-based SAEA, MCEA/D. It builds classification models in low-dimensional spaces and then utilizes these models to estimate good solutions without expensive function evaluations. Experimental results statistically confirm that the proposed algorithm derives state of-the-art performance in many experimental cases.