Thermoelectric properties as well as electronic and magnetic properties of Heusler alloy $$\hbox {Cr}_{2}\hbox {ZnSi}$$ are investigated by employing the first-principles calculations in conjunction with the Boltzmann transport theory and deformation potential (DP) theory. The system is confirmed to be a fully compensated ferrimagnetic spin-gapless semiconductor. We obtain optimized lattice constant of 5.846 A and the zero net magnetic moment. The calculated band structure, served as a hint for its promising thermoelectric properties, shows a zero-width energy gap in the spin-up direction together with an open energy gap in the spin-down one. A detailed study of the chemical potential and temperature dependence of the Seebeck coefficient, lattice and electronic thermal conductivities and hence the figure of merit (ZT) is carried out. The n-type system shows higher ZT values than p-type one in both spin directions, indicating the better thermoelectric performance of n-type system for thermoelectric applications.