We present a three-dimensional (3D) dynamical model for the broad [Fe iii] emission observed in η Carinae using the Hubble Space Telescope/Space Telescope Imaging Spectrograph (STIS). This model is based on full 3D smoothed particle hydrodynamics simulations of η Car’s binary colliding winds. Radiative transfer codes are used to generate synthetic spectroimages of [Fe iii] emission-line structures at various observed orbital phases and STIS slit position angles (PAs). Through a parameter study that varies the orbital inclination i, the PA θ that the orbital plane projection of the line of sight makes with the apastron side of the semimajor axis and the PA on the sky of the orbital axis, we are able, for the first time, to tightly constrain the absolute 3D orientation of the binary orbit. To simultaneously reproduce the blueshifted emission arcs observed at orbital phase 0.976, STIS slit PA =+38° and the temporal variations in emission seen at negative slit PAs, the binary needs to have an i≈ 130° to 145°, θ≈−15° to +30° and an orbital axis projected on the sky at a PA ≈ 302° to 327° east of north. This represents a system with an orbital axis that is closely aligned with the inferred polar axis of the Homunculus nebula, in 3D. The companion star, ηB, thus orbits clockwise on the sky and is on the observer’s side of the system at apastron. This orientation has important implications for theories for the formation of the Homunculus and helps lay the groundwork for orbital modelling to determine the stellar masses.