A thermophotovoltaic (TPV) energy converter harnesses thermal photons emitted by a hot body and converts them to electricity. When the radiative heat exchange between the emitter and PV cell is spectrally monochromatic, the TPV system can approach the Carnot thermodynamic efficiency limit. Nonetheless, this occurs at the expense of vanishing extracted electrical power density. Conversely, a spectrally broadband radiative heat exchange between the emitter and the cell yields maximal TPV power density at the expense of low efficiency. By leveraging hot-carriers as a means to mitigate thermalization losses within the cell, we demonstrate that one can alleviate this trade-off between power density and efficiency. Via detailed balance analysis, we show analytically that one can reach near-Carnot conversion efficiencies close to the maximum power point, which is unattainable with conventional TPV systems. We derive analytical relations between intrinsic device parameters and performance metrics, which serve as design rules for hot-carrier-based TPV systems.