Leibniz algebras have been increasingly used in gauging procedures in supergravity, as it can be shown that embedding tensors naturally induce such algebraic structures. Recent mathematical interpretations of these gauging procedures involve what are called 'Lie-Leibniz triples', of which differential crossed modules are particular cases. This paper is devoted to presenting a new construction of tensor hierarchies from this point of view, that we deem clearer and more straightforward than previous derivations. Moreover, this new construction is a significant improvement since it enables us to draw out a relationship between tensor hierarchies and differential crossed modules. We show in particular that there exists a faithful injective-on-objects functor from semi-strict Lie-Leibniz triples to differential graded Lie algebras (dgLa), whose restriction to differential crossed modules is the canonical assignment associating to any differential crossed module its corresponding unique 3-term differential graded Lie algebra. We stress that such a functorial construction suggests the existence of further well-defined Leibniz gauge theories.