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How subduction initiates with mechanically unfavorable lithospheric heterogeneities is important and rarely studied. We investigate this with a geodynamic model for the Puysegur Incipient Subduction Zone (PISZ) south of New Zealand. The model incorporates a true free surface, elasto‐visco‐plastic rheology and phase changes. Our predictions fit the morphology of the Puysegur Trench and Ridge and the deformation history on the overriding plate. We show how a new thrust fault forms and evolves into a smooth subduction interface, and how a preexisting weak zone can become a vertical fault inboard of the thrust fault during subduction initiation, consistent with two‐fault system at PISZ. The model suggests that the PISZ may not yet be self‐sustaining. We propose that the Snares Zone (or Snares Trough) is caused by plate coupling differences between shallower and deeper parts, the tectonic sliver between two faults experiences strong rotation, and low‐density material accumulates beneath the Snares Zone.