In this paper, we propose an MPC-based precision cooling strategy (PCS) for energy efficient thermal management of automotive air conditioning (A/C) system. The proposed PCS is able to provide precise tracking of the time-varying cooling power trajectory, which is assumed to match the passenger comfort requirements. In addition, by leveraging the emerging connected and automated vehicles (CAVs) technology, vehicle speed preview can be incorporated in our A/C thermal management strategy for further energy efficiency improvement. This proposed A/C thermal management strategy is developed and evaluated based on a physics-based A/C system model (ACSim) from Ford Motor Company for the vehicles with electrified powertrains. In a comparison with Ford benchmark case over SC03 cycle, for tracking the same cooling power trajectory, the proposed PCS provides 4.9% energy saving at the cost of a slight increase in the cabin temperature (less than 1$^oC$). It is also demonstrated that by coordinating with future vehicle speed and shifting the A/C power load, the A/C energy consumption can be further reduced.