The transverse-field Ising model on the Sierpiński fractal, which is characterized by the fractal dimension log₂3≈1.585, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field h_c=1.865. The critical exponents β≈0.198 and δ≈8.7 are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.