In this paper, we introduce a new kind of graph energy, the minimum edge covering energy, $ E_{C_E}(G) $. It depends both on the underlying graph $ G $, and on its particular minimum edge covering $ C_E $. Upper and lower bounds for $ E_{C_E}(G) $ are established. The minimum edge covering energy and some of the coefficients of the polynomial of well-known families of graphs like Star, Path and Cycle Graphs are computed.