The latest High Efficiency Video Coding (HEVC) standard significantly improves coding efficiency over its previous video coding standards. The expense of such improvement is enormous computational complexity, from both encoding and decoding sides. Since computational capability and power capacity are diverse across portable devices, it is necessary to reduce decoding complexity to a target with tolerable quality loss, so called complexity control. This paper proposes a Saliency-Guided Complexity Control (SGCC) approach for HEVC decoding, which reduces the decoding complexity to the target with minimal perceptual quality loss. First, we establish the SGCC formulation to minimize perceptual quality loss at the constraint on reduced decoding complexity, which is achieved via disabling Deblocking Filter (DF) and simplifying Motion Compensation (MC) of some non-salient Coding Tree Units (CTUs). One important component in this formulation is the modelled relationship between decoding complexity reduction and DF disabling/MC simplification, which determines the control accuracy of our approach. Another component is the modelled relationship between quality loss and DF disabling/MC simplification, responsible for optimizing perceptual quality. By solving the SGCC formulation for a given target complexity, we can obtain the DF and MC settings of each CTU, and then decoding complexity can be reduced to the target. Finally, the experimental results validate the effectiveness of our SGCC approach, from the aspects of control performance, complexity-distortion performance, fluctuation of quality loss and subjective quality.