We consider an online version of the well-studied network utility maximization problem, where users arrive one by one and an operator makes irrevocable decisions for each user without knowing the details of future arrivals. We propose a threshold-based algorithm and analyze its worst-case performance. We prove that the competitive ratio of the proposed algorithm is linearly increasing in the number of links in a network and show this competitive analysis is tight. Extensive trace-driven simulations are conducted to demonstrate the performance of our proposed algorithm. In addition, since worst-case scenarios rarely occur in practice, we devise an adaptive implementation of our algorithm to improve its average-case performance and validate its effectiveness via simulations.