The preparation of stable metal–organic framework tubular membranes for liquid mixture separation has always been a tough problem. Herein, a flexibly prepared, continuous, and stable Al-based CAU-10 membrane with ceramic tubes as support was developed by a thin γ-Al2O3 layer induction method and employed for the first time in the separation of MeOH/MTBE azeotropic mixtures through pervaporation. The γ-Al2O3 layer introduced on the tube by the dip-coating technique not only facilitated the crystal nucleation and continuous growth of CAU-10 membranes but also reinforced the membrane stability by acting as anchoring sites between the membrane and the substrate. The as-prepared CAU-10 tubular membrane exhibited excellent separation performance for MeOH/MTBE mixtures owing to its suitable pore size and good hydrophilicity. For the separation of 14.3 wt % MeOH/MTBE azeotropic mixtures, the membrane displayed a permeation flux of 1.76 kg/(m2 h) and a high separation factor of 2041 at 50 °C, which is better than most of the membranes reported for the pervaporation separation of MeOH/MTBE mixtures. More excitedly, the CAU-10 tubular membrane continued to be tested for more than 7 days without any obvious change in pervaporation performance, demonstrating its outstanding operational and hydrothermal stability. Accordingly, the CAU-10 tubular membrane would be a promising alternative to separate the MeOH/MTBE azeotropic mixtures due to its easy preparation, high performance, and excellent stability.