Elastic strain has the potential for a controlled manipulation of the band gap and spin-polarized Dirac states of topological materials, which can lead to pseudomagnetic field effects, helical flat bands, and topological phase transitions. However, practical realization of these exotic phenomena is challenging and yet to be achieved. Here we show that the Dirac surface states of the topological insulator Bi₂Se₃ can be reversibly tuned by an externally applied elastic strain. Performing in situ X-ray diffraction and in situ angle-resolved photoemission spectroscopy measurements during tensile testing of epitaxial Bi₂Se₃ films bonded onto a flexible substrate, we demonstrate elastic strains of up to 2.1% and quantify the resulting changes in the topological surface state. Our study establishes the functional relationship between the lattice and electronic structures of Bi₂Se₃ and, more generally, demonstrates a new route toward momentum-resolved mapping of strain-induced band structure changes.