The NOvA long-baseline neutrino experiment uses a pair of large, segmented, liquid-scintillator calorimeters to study neutrino oscillations, using GeV-scale neutrinos from the Fermilab NuMI beam. These detectors are also sensitive to the flux of neutrinos which are emitted during a core-collapse supernova through inverse beta decay interactions on carbon at energies of O(10 MeV). This signature provides a means to study the dominant mode of energy release for a core-collapse supernova occurring in our galaxy. We describe the data-driven software trigger system developed and employed by the NOvA experiment to identify and record neutrino data from nearby galactic supernovae. This technique has been used by NOvA to self-trigger on potential core-collapse supernovae in our galaxy, with an estimated sensitivity reaching out to 10 kpc distance while achieving a detection efficiency of 23% to 49% for supernovae from progenitor stars with masses of 9.6 M-circle dot to 27 M-circle dot, respectively.