The conventional approach to high-precision narrow-angle astrometry using a long baseline interferometer is to directly measure the fringe packet separation of a target and a nearby reference star. This is done by means of a technique known as phase-referencing which requires a network of dual beam combiners and laser metrology systems. Using an alternative approach that does not rely on phase-referencing, the narrow-angle astrometry of several closed binary stars (with separation less than 2″), as described in this paper, was carried out by observing the fringe packet crossing event of the binary systems. Such an event occurs twice every sidereal day when the line joining the two stars of the binary is is perpendicular to the projected baseline of the interferometer. Observation of these events is well suited for an interferometer in Antarctica. Proof of concept observations were carried out at the Sydney University Stellar Interferometer (SUSI) with targets selected according to its geographical location. Narrow-angle astrometry using this indirect approach has achieved sub-100 micro-arcsecond precision.