We present the discovery of eclipses in the X-ray light curves of the X-ray binary Swift J1858.6-0814. From these, we find an orbital period of $P=76841.3_{-1.4}^{+1.3}$ s (≍21.3 h) and an eclipse duration of $t_{\rm ec}=4098_{-18}^{+17}$ s (≍1.14 h). We also find several absorption dips during the pre-eclipse phase. From the eclipse duration to orbital period ratio, the inclination of the binary orbit is constrained to i > 70°. The most likely range for the companion mass suggests that the inclination is likely to be closer to this value than 90. The eclipses are also consistent with earlier data, in which strong variability ('flares') and the long orbital period prevent clear detection of the period or eclipses. We also find that the bright flares occurred preferentially in the post-eclipse phase of the orbit, likely due to increased thickness at the disc-accretion stream interface preventing flares being visible during the pre-eclipse phase. This supports the notion that variable obscuration is responsible for the unusually strong variability in Swift J1858.6-0814.