We introduce a distributed algorithm, termed noise-robust distributed maximum consensus (RD-MC), for estimating the maximum value within a multi-agent network in the presence of noisy communication links. Our approach entails redefining the maximum consensus problem as a distributed optimization problem, allowing a solution using the alternating direction method of multipliers. Unlike existing algorithms that rely on multiple sets of noise-corrupted estimates, RD-MC employs a single set, enhancing both robustness and efficiency. To further mitigate the effects of link noise and improve robustness, we apply moving averaging to the local estimates. Through extensive simulations, we demonstrate that RD-MC is significantly more robust to communication link noise compared to existing maximum-consensus algorithms.