We develop an algorithm for the axiomatic justification problem in social choice that is sufficiently efficient to be applicable in decision making scenarios of real practical interest. Given a profile of individual preferences, a suggested election outcome, and a corpus of axioms encoding fundamental normative principles of electoral fairness, solving this justification problem involves computing a minimal set of instances of some of the axioms in the corpus that together rule out any outcome that is different from the one we want to justify. Our approach combines the use of state-of-the-art tools for computing minimally unsatisfiable sets of constraints with a graph-search algorithm. The latter searches the graph induced by the set of all axiom instances in an incremental manner and relies on a number of heuristics to further improve performance.