We present near-infrared spectra of 10 planetary nebulae (PNe) in the Large and Small Magellanic Clouds (LMC and SMC), acquired with the FIRE and GNIRS spectrometers on the 6.5 m Baade and 8.1 m Gemini South Telescopes, respectively. We detect Se and/or Kr emission lines in eight of these objects, the first detections of n-capture elements in Magellanic Cloud PNe. Our abundance analysis shows large s-process enrichments of Kr (0.6–1.3 dex) in the six PNe in which it was detected, and Se is enriched by 0.5–0.9 dex in five objects. We also estimate upper limits to Rb and Cd abundances in these objects. Our abundance results for the LMC are consistent with the hypothesis that PNe with 2–3 M ⊙ progenitors dominate the bright end of the PN luminosity function in young gas-rich galaxies. We find no significant correlations between s-process enrichments and other elemental abundances, central star temperature, or progenitor mass, though this is likely due to our small sample size. We determine S abundances from our spectra and find that [S/H] agrees with [Ar/H] to within 0.2 dex for most objects, but is lower than [O/H] by 0.2–0.4 dex in some PNe, possibly due to O enrichment via third dredge-up. Our results demonstrate that n-capture elements can be detected in PNe belonging to nearby galaxies with ground-based telescopes, allowing s-process enrichments to be studied in PN populations with well-determined distances. This paper includes data obtained with the 6.5-m Magellan Telescopes located at Las Campanas Observatory, Chile, and with the Gemini-South Telescope at Cerro Pachon, Chile.