We analyze the accuracy of collective decision-making in socially connected populations, where agents update binary choices through local interactions on a network. Each agent receives a private signal that is biased -- even marginally -- toward the correct alternative, and social influence mediates the aggregation of these signals. We show analytically that, in the large-population limit, the probability of a correct majority converges to a nontrivial expression involving the regularized incomplete beta function. Remarkably, this collective accuracy surpasses that of any individual agent whenever private signals are better than random, revealing that network-mediated influence can enhance, rather than impair, group performance. Our findings may inform the design of resilient decision-making systems in social, biological, and engineered networks, where accuracy must emerge from interdependent and noisy agents.