Human mobility connects populations and can lead to large fluctuations in population density, both of which are important drivers of epidemics. Measuring population mobility during infectious disease outbreaks is challenging, but is a particularly important goal in the context of rapidly growing and highly connected urban centers in low and middle income countries, which can act to amplify and spread local epidemics nationally and internationally. Here, we combine estimates of population movement from mobile phone data for over 4 million subscribers in the megacity of Dhaka, Bangladesh, one of the most densely populated cities globally. We combine mobility data with epidemiological data from a household survey, to understand the role of population mobility on the spatial spread of the mosquito-borne virus chikungunya within and outside Dhaka city during a large outbreak in 2017. The peak of the 2017 chikungunya outbreak in Dhaka coincided with the annual Eid holidays, during which large numbers of people traveled from Dhaka to their native region in other parts of the country. We show that regular population fluxes around Dhaka city played a significant role in determining disease risk, and that travel during Eid was crucial to the spread of the infection to the rest of the country. Our results highlight the impact of large- scale population movements, for example during holidays, on the spread of infectious diseases. These dynamics are difficult to capture using traditional approaches, and we compare our results to a standard diffusion model, to highlight the value of real-time data from mobile phones for outbreak analysis, forecasting, and surveillance.