Given a sequence of random graphs, we address the problem of online monitoring and detection of changes in the underlying data distribution. To this end, we adopt the Random Dot Product Graph (RDPG) model which postulates each node has an associated latent vector, and inner products between these vectors dictate the edge formation probabilities. Existing approaches for graph change-point detection (CPD) rely either on extensive computation, or they store and process the entire observed time series. In this paper we consider the cumulative sum of a judicious monitoring function, which quantifies the discrepancy between the streaming graph observations and the nominal model. This reference distribution is inferred via spectral embeddings of the first few graphs in the sequence, and the monitoring function can be updated in an efficient, online fashion. We characterize the distribution of this running statistic, allowing us to select appropriate thresholding parameters that guarantee error-rate control. The end result is a lightweight online CPD algorithm, with a proven capability to flag distribution shifts in the arriving graphs. The novel method is tested on both synthetic and real network data, corroborating its effectiveness in quickly detecting changes in the input graph sequence.