With the advent of online networks, societies are substantially more connected with individual members able to easily modify and maintain their own social links. Here, we show that active network maintenance exposes agents to confirmation bias, the tendency to confirm one's beliefs, and we explore how this affects collective opinion formation. We introduce a model of binary opinion dynamics on a complex network with fast, stochastic rewiring and show that confirmation bias induces a segregation of individuals with different opinions. We use the dynamics of global opinion to generally categorize opinion update rules and find that confirmation bias always stabilizes the consensus state. Finally, we show that the time to reach consensus has a non-monotonic dependence on the magnitude of the bias, suggesting a novel avenue for large-scale opinion engineering.