Numerous studies on climate change and variability have revealed that these phenomena have noticeable influence on the epidemiology of dengue fever, and such relationships are complex due to the role of the vector—the Aedes mosquitoes. By undertaking a step-by-step approach, the present study examined the effects of climatic factors on vector abundance and subsequent effects on dengue cases of Dhaka city, Bangladesh. Here, we first analyzed the time-series of Stegomyia indices for Aedes mosquitoes in relation to temperature, rainfall and relative humidity for 2002–2013, and then in relation to reported dengue cases in Dhaka. These data were analyzed at three sequential stages using the generalized linear model (GLM) and generalized additive model (GAM). Results revealed strong evidence that an increase in Aedes abundance is associated with the rise in temperature, relative humidity, and rainfall during the monsoon months, that turns into subsequent increases in dengue incidence. Further we found that (i) the mean rainfall and the lag mean rainfall were significantly related to Container Index, and (ii) the Breteau Index was significantly related to the mean relative humidity and mean rainfall. The relationships of dengue cases with Stegomyia indices and with the mean relative humidity, and the lag mean rainfall were highly significant. In examining longitudinal (2001–2013) data, we found significant evidence of time lag between mean rainfall and dengue cases.