We report the detection of the unidentified emission feature at 21 microns in infrared spectra of eight post-AGB objects in the Magellanic Clouds. This is the first detection of the feature in extragalactic objects. The observations were made by the IRS instrument on the Spitzer Space Telescope, as part of multiple observing programs. The eight new detections are considerable, given that only 15 post-AGB objects in the Galaxy have been identified with 21-micron features. The infrared spectra with 21-micron features in the Magellanic Clouds tend to show stronger emission from the accompanying aromatic hydrocarbon (PAH) features than those in the Galaxy do. The dust in the new objects also generally appears to be hotter, which might imply that they have not evolved as far from the AGB as their Galactic analogues. The 21-micron features are weaker on average than in the Galactic objects, although two of the objects do show strong features. We present the Spitzer spectra of these objects as well as the available ground-based optical and near-infrared photometry and some preliminary results from radiative transfer models.