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We show that the observed time-reversal symmetry breaking (TRSB) of the superconducting state in Sr₂RuO₄ can be understood as originating from inhomogeneous strain fields near edge dislocations of the crystal. Specifically, we argue that, without strain inhomogeneities, Sr₂RuO₄ is a single-component, time-reversal symmetric superconductor, likely with d_x²-y² symmetry. However, due to the strong strain inhomogeneities generated by dislocations, a slowly-decaying sub-leading pairing state contributes to the condensate in significant portions of the sample. As it phase winds around the dislocation, time-reversal symmetry is locally broken. Global phase locking and TRSB occur at a sharp Ising transition that is not accompanied by a change of the single-particle gap and yields a very small heat capacity anomaly. Our model thus explains the puzzling absence of a measurable heat capacity anomaly at the TRSB transition in strained samples, and the dilute nature of the time-reversal symmetry broken state probed by muon spin rotation experiments.