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Controlling the bandstructure through local-strain engineering is an exciting avenue for tailoring optoelectronic properties of materials at the nanoscale. Atomically thin materials are particularly well-suited for this purpose because they can withstand extreme nonhomogeneous deformations before rupture. Here, we study the effect of large localized strain in the electronic bandstructure of atomically thin MoS₂. Using photoluminescence imaging, we observe a strain-induced reduction of the direct bandgap and funneling of photogenerated excitons toward regions of higher strain. To understand these results, we develop a nonuniform tight-binding model to calculate the electronic properties of MoS₂ nanolayers with complex and realistic local strain geometries, finding good agreement with our experimental results.