This study aims to introduce a new convective vertical mixing scheme rooted in the Eddy-Diffusivity Mass-Flux (EDMF) approach, itself derived from first principles. The integration of the mass-flux (MF) concept with the Eddy-Diffusivity (ED) approach has long been studied and applied in global and regional atmospheric models for parameterizing convection, both dry and cloudy, and sees a growing interest in ocean models. This type of closure involves separating vertical turbulent fluxes into two components: a diffusion term that addresses local small-scale mixing in a near isotropic environment and a mass-flux transport term that accounts for the non-local transport due to vertically coherent plumes within the environment. Here, we exploit the multi-fluid averaging technique underlying the MF concept to propose an original formulation of a scheme that possesses properties not satisfied by most existing EDMF formulations. Consistent energy budgets between resolved and subgrid scales are derived for different multi-component fluids including seawater and dry atmosphere (in Boussinesq and anelastic cases). This guarantees that all mean kinetic, potential, and internal energy sinks and sources as long as turbulent kinetic energy (TKE) transport associated with EDMF terms are exactly added or subtracted to the TKE budget, effectively rectifying energy biases existing in prior EDMF schemes. Notably, this analysis facilitates a clear separation of convective and turbulent small-scale energy reservoirs. In addition to bulk energy transfers, we provide guidelines to avoid spurious energy fluxes at the fluid's boundary when using EDMF. We illustrate the performance of the proposed \textit{energetically consistent EDMF scheme} in the context of oceanic convection. When compared with Large Eddy Simulations (LES) of oceanic convection, our scheme can reproduce mean fields as well as higher-order moments such as TKE, vertical fluxes, and turbulent transport of TKE. The energetic consistency is key to obtaining realistic TKE and turbulent transport of TKE profiles. To further illustrate that the MF concept is a credible alternative to the traditional approaches used in the oceanic context (using an enhanced vertical diffusion or a counter gradient term \textit{à la} KPP) the proposed scheme is validated in a single-column configuration against observational data of oceanic convection from the LION buoy. % Last but not least, during the theoretical development of the scheme, we maintain transparency regarding underlying assumptions and systematically assess their validity in the light of LES data.