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The ever-growing data rate demand from cellular users increases the associated power consumption that directly contributes to the global warming of the environment. Also, achieving high system capacity by increasing the density of the base stations (BSs) increases the number of handovers (HOs) which for moderate-to-high-speed users implies significant signaling traffic overhead. One of the key research objectives of this paper is to identify the different signaling overheads occurring during the HO procedure in current 3GPP cellular networks (e.g. Long Term Evolution (LTE)/ New Radio (NR)) and, among those, which are the main contributors to power consumption. Specifically, we analyze the impact of signaling messages transmitted and received during HO on the power consumption for both the BS and the User Equipment (UE). System-level simulations are performed for a detailed quantitative analysis. Our analysis shows that the transmission of the measurement reports is the largest contributor to air-interface signaling and that its contributed total power consumption is higher than the random access channel (RACH) signaling and the signaling confirming the HO. To eliminate measurement reports and effectively reduce the power consumption associated with the HO in future networks, we propose a HO procedure that exploits uplink (UL) reference signals (RSs), namely the sounding reference signal (SRS), transmitted by UEs. The numerical results show that the proposed SRS-based method reduces the total power consumption during the HO procedure by 30% in comparison to the legacy downlink RS based measurement method in current cellular networks. Also, this method improves the UE battery lifetime by reducing the RS transmissions and measurements significantly, UE transmitted power consumption by 48% and received power consumption by 27%.