Adaptive bitrate (ABR) algorithms aim to make optimal bitrate decisions in dynamically changing network conditions to ensure a high quality of experience (QoE) for the users during video streaming. However, most of the existing ABRs share the limitations of predefined rules and incorrect assumptions about streaming parameters. They also come short to consider the perceived quality in their QoE model, target higher bitrates regardless, and ignore the corresponding energy consumption. This joint approach results in additional energy consumption and becomes a burden, especially for mobile device users. This paper proposes GreenABR, a new deep reinforcement learning-based ABR scheme that optimizes the energy consumption during video streaming without sacrificing the user QoE. GreenABR employs a standard perceived quality metric, VMAF, and real power measurements collected through a streaming application. GreenABR's deep reinforcement learning model makes no assumptions about the streaming environment and learns how to adapt to the dynamically changing conditions in a wide range of real network scenarios. GreenABR outperforms the existing state-of-the-art ABR algorithms by saving up to 57% in streaming energy consumption and 60% in data consumption while achieving up to 22% more perceptual QoE due to up to 84% less rebuffering time and near-zero capacity violations.