We study the Kibble–Zurek mechanism in the transverse Ising chain coupled to a dissipative boson bath, using a new numerical method with the infinite time evolving block decimation combined with the discrete-time path integral. We first show the ground-state phase diagram and confirm that a quantum phase transition takes place in the presence of system-bath coupling. Then we present the time dependence of the energy expectation value of the spin Hamiltonian and the scaling of the kink density with respect to the time period over which the spin Hamiltonian crosses a quantum phase transition. The energy of spins starts to grow from the energy at the ground state of the full system near a quantum phase transition. The kink density decays in accordance with a power law with respect to the time period. These results confirm that the Kibble–Zurek mechanism occurs. We discuss the exponent for the decay of the kink density in comparison with a theoretical result with the quantum Monte Carlo simulation. A comparison with an experimental study is also briefly mentioned.