We present the first systematic measurement of the tilt of the pion emission source in relativistic Au+Au collisions at center-of-mass energies per nucleon pair, $\sqrt{s_{NN}}$ = 7.7, 14.5, 17.3 and 27 GeV, using data from the STAR experiment. The tilt angle and final freeze-out eccentricity are extracted through azimuthally sensitive femtoscopy of identical pion pairs. Our results reveal a strong dependence of the tilt parameter on the pair transverse momentum, indicating that the apparent source geometry is strongly coupled to expansion dynamics. Moreover, we observe a rapid decrease of the tilt magnitude with increasing collision energy, consistent with the emission source approaching longitudinal boost invariance at higher energies. These findings demonstrate that the commonly assumed boost-invariant geometry is insufficient and highlight the necessity of exploring the spatial structure of a tilted source, which is required in hydrodynamic models to reproduce features of the longitudinally expanding system, such as the slope of the directed flow. Comparisons with the UrQMD transport model show that it reproduces the overall energy dependence of the tilt magnitude qualitatively, but not quantitatively.