The problem of mixed convection in a differentially heated lid-driven square cavity filled with Cu-water nanofluid under effect of a magnetic field is investigated numerically. The left and right walls of the cavity are kept at temperatures of $T_h$ and $T_c$ respectively while the horizontal walls are adiabatic. The top wall of the cavity moves in own plane from left to right. The effects of some pertinent parameters such as Richardson number (ranging from 0.1 to 10), the volume fraction of the nanoparticles (ranging 0 to 0.1) and the Hartmann number (ranging from 0 to 60) on the fluid flow and temperature fields and the rate of heat transfer in the cavity are investigated. It must be noted that in all calculations the Prandtl number of water as the pure fluid is kept at 6.8, while the Grashof number is considered fixed at 104. The obtained results show that the rate of heat transfer increases with an increase of the Reynolds number, while but it decreases with increase in the Hartmann number. Moreover it is found that based the Richardson and Hartmann numbers by increase in volume fraction of the nanoparticles the rate of heat transfer can be enhanced or deteriorated compared to the based fluid.