It is well known that transport in lightly n-doped SrTiO₃ involves light and heavy electron bands. We have found that upon application of moderate quasi-isotropic pressures, the relative positions of these subbands are changed by a few meV and, eventually, a band inversion occurs at ~1 kbar. Such effects are, however, suppressed in the closely related KTaO₃ perovskite. We show that the extremely subtle electronic reconfiguration in SrTiO₃ is triggered by strain-induced structural transformations that are accompanied by remarkable mobility enhancements up to about Δμ/μ≈300%. Our results provide a microscopic rationale for the recently discovered transport enhancement under strain and underscore the role of the internal structural degrees of freedom in the modulation of the perovskite electronic properties.