Accurate first-principles predictions of the structural, electronic, magnetic, and electrochemical properties of cathode materials can be key in the design of novel efficient Li-ion batteries. Spinel-type cathode materials LixMn2O4 and LixMn1.5Ni0.5O4 are promising candidates for Li-ion battery technologies, but they present serious challenges when it comes to their first-principles modeling. Here, we use density-functional theory with extended Hubbard functionals - DFT+U+V with on-site U and inter-site V Hubbard interactions - to study the properties of these transition-metal oxides. The Hubbard parameters are computed from first-principles using density-functional perturbation theory. We show that while U is crucial to obtain the right trends in properties of these materials, V is essential for a quantitative description of the structural and electronic properties, as well as the Li-intercalation voltages. This work paves the way for reliable first-principles studies of other families of cathode materials without relying on empirical fitting or calibration procedures.