  One approach to probe the still controversial superconductivity in
Sr$_2$RuO$_4$ is to apply external perturbations that break the underlying
tetragonal crystalline symmetry. Chiral $p_x+ip_y$ and $d_{xz}+id_{yz}$ states
respond to such perturbations in ways that may help to distinguish them from
other superconducting pairings. However, past experimental efforts along this
line, using uniaxial strains and magnetic fields parallel to the RuO$_2$ plane,
have not been able to reach an unambiguous conclusion. In this study, we
propose to further examine the possibility of chiral superconducting order in
Sr$_2$RuO$_4$ using an alternative tetragonal-symmetry-breaking perturbation --
in-plane supercurrent. We study the superconducting phase diagram as a function
of both temperature and the applied supercurrent. Supercurrent generically
splits the transition of the two chiral order parameter components, and we show
that the splitting can give rise to visible specific heat anomalies.
Furthermore, supercurrent parallel and anti-parallel to the unidirectional
propagation of the chiral edge modes impact the edge states in different
manner. This difference manifests in the tunneling spectrum, thereby providing
an additional means to probe the chirality even when the related spontaneous
edge current is vanishingly small. Finally, we discuss the distinction of
supercurrent responses in non-chiral time-reversal-symmetry-breaking
superconducting states. Our proposal can be applied to other candidate chiral
superconductors.
