The question whether energy is conserved in an expanding universe remains one of the central conceptual issues of relativistic cosmology. The absence of a global timelike Killing vector and the lack of a local gravitational energy density often lead to the conclusion that cosmological redshift represents a genuine violation of energy conservation. In this work we reconsider this problem within the framework of Brown-York quasi-local energy and its associated balance law. For comoving spherical boundaries in FLRW spacetimes we show that the ordinary stress-energy flux through the boundary vanishes identically for arbitrary perfect fluids, while the unreferenced Brown-York energy evolves entirely due to a purely geometrical boundary contribution. We further demonstrate that the timelike boundary world tube possesses a natural dynamical reference geometry. The corresponding reference subtraction removes the universal background contribution of the unreferenced Brown-York energy and leads to a quasi-local energy whose leading nontrivial term exactly reproduces the enclosed matter energy. For radiation dominated cosmologies, the resulting expansion is $E_{\rm BY} = H^2R^3 /2 - H^4R^5 / 8 + \mathcal{O}(R^7)$, where the leading term coincides precisely with the energy of the enclosed radiation responsible for cosmological redshift. Higher-order contributions naturally possess the structure expected for gravitational self-energy corrections. These results suggest that cosmological redshift can be understood within a fully quasi-local conservation framework. The apparent loss of photon energy therefore need not indicate a breakdown of energy conservation but may instead reflect the fundamentally quasi-local character of gravitational energy in general relativity.