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.